Dipper-James 22 7

Transcription

1 22 7

2 Frobenius GL n (E) Φ d -torus Young Tits GL n (E) GL n (q) Harish-Chandra Harish-Chandra Mackey Howlett-Lehrer Harish-Chandra Hall GL n(q) Hall Hall-Littlewood Hall Deligne-Lusztig A Hecke Hall Green

3 ii 4.5 Deligne-Lusztig cuspidal Brauer GL n (q) χ n,r GL n (q) cuspidal A Hecke Harish-Chandra Hecke Dipper Howlett-Lehrer Hecke Specht A (1) e 1 Kac-Moody Fock A Hecke F GL n (q)

4 1 1.1 G = GL n (C) G C ω (g, h) G G gh G g G g 1 G C ω C ω Lie n GL n (C) Lie Lie C ω E affine E- E E- R X = Spec R E X(E) = Hom E-alg (R, E) affine G = Spec R G R Hopf E-Hopf GL n (E) 1.1 L L- H Hopf

5 2 1 L- : H H L H L- ε : H L S : H H (1) ( Id) = (Id ) (2) (ε Id) = (Id ε) = Id (3) (S Id) = (Id S) = η ε η : L H λ λ 1 H H L H L- (h 1 h 1)(h 2 h 2) = h 1 h 2 h 1h 2 h H (h) = n (1) (2) (3) n i=1 n i=1 n i=1 (h (1) i ) h (2) i = n h (1) i=1 i=1 i (h (2) i ) H L H L H ε(h (1) i )h (2) i = n ε(h (2) i )h (1) i = h i=1 S(h (1) i )h (2) i = n h (1) i S(h (2) i ) = ε(h)1 H i=1 h (1) i h (2) i coproduct ε counit S antipode antipode E E[GL n ] := E[{X ij } 1 i,j n, det X ] ij X ij X = (X ij ) 1 i,j n (i, j)- X i j D ij ij = ( 1) i+j D ji

6 1.1 3 (X ij ) = n k=1 X ik X kj, ε(x ij ) = δ ij, S(X ij ) = 1 det X ij E[GL n ] Hopf σ S n γ(σ) = {(i, j) i < j, σ(i) > σ(j)} sgn(σ) = ( 1) γ(σ) S n sgn : S n {±1} det X = sgn(σ)x 1σ(1) X nσ(n) σ S n n (X ij ) = X ik X kj k=1 : E[{X ij } 1 i,j n ] E[GL n ] E E[GL n ] (det X) = n k 1,,k n =1 X 1k1 X nkn σ S n sgn(σ)x k1σ(1) X knσ(n) X k11 X k1n sgn(σ)x k1σ(1) X knnσ(n) = σ S n X kn1 X knn X k 1,, k n k 1,, k n τ S n k i = τ(i) (1 i n) σ S n sgn(σ)x k1σ(1) X knnσ(n) = sgn(τ) det X

7 4 1 (det X) = τ S n sgn(τ)x 1τ(1) X nτ(n) det X = det X det X. 1 ( det X ) = 1 det X 1 det X : E[GL n ] E[GL n ] E[GL n ] well-defined 1 ε( det X ) = 1 ε : E[GL n] E well-defined. S(X ij ) Y := X 1 (i, j) S(det X) = sgn(σ)y nσ(n) Y 1σ(1) = det Y = 1 σ S n det X. 1 S( ) = det X det X S : E[GL n] E[GL n ] well-defined E[GL n ] Hopf h = X ij (1)(2)(3) (1)(2) (3) 1.2 L L- H Hopf R L- G(R) := Hom L-alg (H, R) ϕψ := (ϕ ψ) (ϕ, ψ G(R)) ε : H L h ε(h)1 R H R ε G(R) ϕ G(R) ϕε = (ϕ ε) = ϕ (Id ε ) = ϕ εϕ = ϕ (ϕ 1 ϕ 2 )ϕ 3 = ((ϕ 1 ϕ 2 ) ϕ 3 )

8 1.1 5 = (ϕ 1 ϕ 2 ϕ 3 ) ( Id) = (ϕ 1 ϕ 2 ϕ 3 ) (Id ) = (ϕ 1 (ϕ 2 ϕ 3 ) ) = ϕ 1 (ϕ 2 ϕ 3 ). ϕ G(R) ϕ 1 H S H ϕ R ϕ L- R ϕ 1 (h 1 h 2 ) = ϕ(s(h 2 )S(h 1 )) = ϕ(s(h 2 ))ϕ(s(h 1 )) = ϕ 1 (h 2 )ϕ 1 (h 1 ) = ϕ 1 (h 1 )ϕ 1 (h 2 ) ϕ 1 G(R) ϕ ϕ : H H R ϕ H ϕϕ 1 = (ϕ ϕ 1 ) = (ϕ ϕ) (Id S) = ϕ((id S) ) = ϕ(η ε) = ε : H L R. ϕ 1 ϕ = ε 1.3 R E- ϕ GL n (R) = Hom E-alg (E[GL n ], R) (ϕ(x ij )) 1 i,j n GL n (R) R n GL n (R) Φ(ϕ) := (ϕ(x ij )) 1 i,j n Φ(ϕψ) ij = ϕψ(x ij ) = (ϕ ψ) (X ij ) = n k=1 Φ(ϕ) ik Φ(ψ) kj Φ(ϕψ) = Φ(ϕ)Φ(ψ) Φ(ε) ϕϕ 1 = ε Φ(ϕ) Φ GL n (R) R

9 6 1 n R n C ϕ(x ij ) = c ij Φ(ϕ) = C ϕ GL n (R) 1.2 E E- E[G] Hopf G(E) = Hom E-alg (E[G], E) E E[G] G(E) Hilbert G(E) f E[G] G(E) 0 f = 0 f = 0 E[G] reduced 0 E[G] reduced E[GL n ] reduced E[G] E[G]/ 0 G(E) E[G] reduced E[G] reduced f E[G] G(E) g G(E) g(f) f(g) (f) = f (1) i f (2) i f(g 1 g 2 ) = f (1) i (g 1 )f (2) i (g 2 ) f(g 1 ) = S(f)(g) G(E) 1 G ε(f) = f(1 G ) 1.2 Hopf coproduct, counit, antipode Hopf 1.4 G 1 (E), G 2 (E) Hopf ϕ : E[G 2 ] E[G 1 ] g g ϕ

10 1.1 7 ϕ : G 1 (E) G 2 (E) f E[G 2 ] f(ϕ(gh)) = (g h) 1 ϕ (f) = (g h) ϕ 2 (f) = (g ϕ ) (h ϕ ) 2 (f) = f(ϕ(g)ϕ(h)) ϕ(gh) = ϕ(g)ϕ(h) f(ϕ(g 1 )) = g S 1 ϕ (f) = g ϕ S 2 (f) = ϕ(g) S 2 (f) = f(ϕ(g) 1 ) ϕ(g 1 ) = ϕ(g) 1 1 G1 = ε 1, 1 G2 = ε 2 f(ϕ(1 G1 )) = ε 1 ϕ (f) = ε 2 (f) = f(1 G2 ) ϕ 1.3 GL n (E) = Hom E-alg (E[GL n ], E) E Hopf Spec E[GL n ] E G E[G]- Abel E[GL n ] E[GL n ] E- E q [GL n ] 1.2 E[GL n ] E q [GL n ] E GL n (E) E[GL n ] Hopf U(gl n ) E q [GL n ] E[GL n ] X ij r 0 r

11 8 1 E[GL n ](r) (E[GL n ](r)) E[GL n ](r) E[GL n ](r) S(n, r) := Hom E (E[GL n ](r), E) E S(n, r) Schur Schur U(gl n ) E q [GL n ] Schur q-schur gl n q-schur Lie GL n (F q ) q-schur q l q 1 q-schur Fock Cherednik O 1.2 Frobenius 1.4 E- H Hopf H F q - H 0 (1) E- H = H 0 Fq E (2) (H 0 ) H 0 Fq H 0 (3) ε(h 0 ) F q (4) S(H 0 ) H 0 H 0 H F q - σ q : H = H 0 Fq E H 0 Fq E = H h 0 λ h 0 λ q G(E) = Hom E-alg (H, E) E- F # : H H F # σ q (h) = h q (h H) F : G(E) G(E)

12 1.2 Frobenius 9 ϕ ϕ F # F F q - H 0 Frobenius F # H 0 q H 0 q H 0 F q - H E- F # σ q (h) = h q F q - H 0 F # det X ] ϕ GL n (E) = Hom E-alg (H, E) H = E[GL n ] H 0 = F q [GL n ] := F q [{X ij } 1 i,j n, H 0 H F q - F (ϕ)(x ij ) = ϕ(x q ij ) = ϕ(x ij) q GL n (E) E n Frobenius q 1.5 E- H Hopf H 0 H F q - H 0 Frobenius coproduct H F # H H E H F # F # H E H H H 0 h H 0 F # (h) = h q F # (h) = (h) q H 0 H 0 (h) = i h (1) i h (2) i (h (1) i, h (2) i H 0 ) (h) q = i h (1) i q (2) q h i (h) q = F # F # ( i h (1) i h (2) i ) = (F # F # ) (h). F # = (F # F # )

13 10 1 F (ϕ)f (ψ) = ((ϕ F # ) (ψ F # )) = (ϕ ψ) (F # F # ) = (ϕ ψ) F # = (ϕψ) F # = F (ϕψ). F GL n (E) Frobenius q Frobenius GL n (E) 1.5 G = GL n (E) G F = {g G F (g) = g} F q GL n (q) GL n (F q ) GL n (q) = {g = (g ij ) 1 i,j n g ij F q, det g 0} GL n (q) Frobenius Frobenius {X q ij X ij} 1 i,j n E[GL n ] I q E[GL n ]/I q Hopf GL n (q) = Hom E-alg (E[GL n ]/I q, E) 1.3 GL n (E) GL n (E) GL n (E)

14 1.3 GL n (E) L L- H Hopf H I Hopf (1) (I) I L H + H L I (2) ε(i) = 0 (3) S(I) I H/I Hopf 1.7 G(E) = Hom E-alg (E[G], E) E E[G] I K Hopf E[K] := E[G]/I K G(E) K(E) := Hom E-alg (E[K], E) G(E) I q = ({X q ij X ij} 1 i,j n ) E[GL n ] Hopf GL n (q) GL n (E) (X q ij X ij) = n k=1 (X q ik X ik) X q kj + n X ik (X q kj X kj) ε(x q ij X ij) = 0 (1), (2) (det X) q det X I q (det X) q (det X) 1 I q S(X q ij X ij) (1) S(X q ij X ij) I q (3) 1.6 (1) E[GL n ] {X ij } 1 j<i n I B k=1 E[GL n ] Hopf (2) E[GL n ] {X ij } 1 j i n I T E[GL n ] Hopf (1) i > j (X ij ) = n k=1 I B I T X ik X kj 1 k < i X ik I B i k n X kj I B

15 12 1 ε(i B ) = 0 (I B ) I B E H + H E I B. S(X ij ) = ij / det X I B D ji I B D ji 1,, i 1, i + 1,, n k 1,, k j 1, k j+1,, k n ±X 1k1 X j 1kj 1 X j+1kj+1 X nkn 1 k 1,, j 1 k j 1, j + 1 k j+1,, n k n i j + 1 k n = n,, k i+1 = i + 1 i k i k i i 1 (2) i j i k k j (I T ) I T E H + H E I T ε(i T ) = 0 D ji 1,, i 1, i+1,, n k 1,, k j 1, k j+1,, k n ±X 1k1 X j 1kj 1 X j+1kj+1 X nkn 1 = k 1,, j 1 = k j 1, j + 1 = k j+1,, n = k n i j 1.8 E[B n ] := E[GL n ]/I B, E[T n ] := E[GL n ]/I T B n := Hom E-alg (E[B n ], E) GL n (E) GL n (E) Borel T n := Hom E-alg (E[T n ], E) GL n (E) GL n (E) torus GL n (E) E B n T n

16 1.4 Φ d -torus 13 B n GL n (E) GL n (E) Borel T n GL n (E) GL n (E) torus 1.1 F F q - F q [GL n ] Frobenius F (B n ) = B n F (T n ) = T n B n (q) := Bn F GL n(q) Borel T n (q) := Tn F GL n(q) torus GL n (E) Borel B F (B) = B B F GL n (q) Borel GL n (E) torus T F (T ) = T T F GL n (q) torus 1.9 GL n (E) P Borel B n P GL n (E) GL n (E) GL n (E) n (e 1,, e n ) {e 1,, e n } E n σ S n g(σ)e i = e σ(i) (1 i n) g(σ) GL n (E) W n = {g(σ) σ S n } σ g(σ) S n N GLn (E)(T n )/T n = W n T n /T n W n W n GL n (E) Weyl Weyl S n W n GL n (E) q S n GL n (q) = GL n (E) F 1.10 GL n (q) P Borel B n (q) P GL n (q) GL n (q) GL n (q) 1.4 Φ d -torus GL n (q) F n q n

17 14 1 GL n (q) = (q n 1)(q n q) (q n q n 1 ) n = q n(n 1) 2 (q i 1) i=1 O n (X) = X n(n 1) 2 n (X i 1) i=1 GL n (q) Φ d (X) 1 d n (X i 1) = Φ d (X) [ n d ] i=1 d a [ n d ] T n GL n (E) S Frobenius F F (S) = S S F = Φ d (q) a n = d a = 1 E 1 q d 1 ζ 1 q d 1 F q d ζ ζ q ζ qd 1 ζ 2 (ζ q ) 2 (ζ qd 1 ) 2 g =.... GL n (E)..... ζ d 1 (ζ qd 1 ) d 1

18 F (g) = g Φ d -torus 15 T = gt n g 1 F (T ) = T t 1 t 2 F : g... g 1 g t q d t q 1... g 1 t d t q d 1 T Frobenius t g t q... t qd 1 g 1 (t F ) q d ϕ(d) {1,, d 1} d Φ d (X) = X ϕ(d) + a 1 X ϕ(d) a ϕ(d) 1 X + 1 (a 1,, a ϕ(d) 1 Z) 1,, d Z/dZ t i t a ϕ(d) 1 i+1 t a 1 i+ϕ(d) 1 t i+ϕ(d) = 1 (i Z/dZ) T S F (S) = S T Frobenius t 1+a ϕ(d) 1q+ +a 1 q ϕ(d) 1 +q ϕ(d) = t Φ d(q) = 1 (t F q d ) S F = Φ d (q)

19 16 1 torus Φ d -torus T GL d (E) Coxeter torus Φ d -torus S S GL n (E) d- Levi GL n (q) 1.2 n = d = 6 T = gt 6 g 1 Coxeter torus Φ 6 (X) = X 2 X + 1 GL 6 (E) Φ 6 -torus t 1 t 2 g... t 6 g 1 (t 1 t 3 = t 2, t 2 t 4 = t 3,, t 5 t 1 = t 6, t 6 t 2 = t 1 ) T t 1,, t 6 t 1 = t 1 5 t 6, t 2 = t 1 5, t 3 = t 1 6, t 4 = t 5 t 1 6 Φ 6 -torus E E Frobenius GL 6 (q) t 5 = t 1 q 6, t q2 q+1 6 = 1 G G G l a G l a G Sylow l e q mod l e = min{k Z 1 q k 1 l } n Z 1 n = n 1 + k 0 n k el k (0 n 1 < e, 0 n k < l)

20 1.4 Φ d -torus 17 (e, l)- 1.8 S 0 GL e (q) Φ e -torus Frobenius Sylow l- l (1) S 0 GL e (q) Sylow l- (2) l- S k GL el k(q) Sylow l- S l k GL el k(q) l GL el k+1(q) i + el k (1 i el k+1 el k ) w(i) = i + el k el k+1 (el k+1 el k + 1 i el k+1 ) S el k+1 w l l S k+1 S k+1 GL el k+1(q) S l k Sylow l- (3) n (e, l)- n = n 1 + n k el k k 0 GL n (q) Sylow l- {1} k 0 S n k k GL n 1 (q) GL el k(q) n k GL n (q) k 0 (1) q 1, q 2 1,, q e 1 1 l GL e (q) Sylow l- Coxeter torus Frobenius F q e Sylow l- q e 1 = Φ k (q) k e Φ k (q) l q k 1 l l Φ e (q) GL e (q) Sylow l- S 0

21 18 1 (2) q k 1 l q mod l k 1 k e k = ek q e 1 l a l b q e = 1 + l a b q k = (1 + l a b) k = 1 + k l a b + k r=2 ( k r ) l ra b r c Z 0 l d k = l c d ( k r ) l ra (r 2) l a+c+1 r l c l 3 ra l c a = a + (l 1)(l c l + 1)a a + 2ca > a + c 0 i < c r l i l k 1 ([ k l k ] r [k l k ] [ r ) l k ] k = i + 1,, c [ k l k ] r [k l k ] [ r ( l k ] = k k l k l k [ r ) l k ] 1 [ r l k ] = 1 l c i i = 0 r 2 ra + c 2a + c > a + c i 1 l 3 l i i + 2 ra + c i l i a i + c (i + 2)a i + c = 2a + (a 1)i + c > a + c l a+c+1

22 1.5 Young 19 k l c q k 1 l a + c N l ν l (N) ν l ( GL n (q) ) = n k=1 ν l (q k 1) = [ n e ] k =1 ν l (q ek 1) = [ n e ] k =1 n (e, l)- ν l ( GL n (q) ) = n n 1 a + n k (l k l + 1) e k 1 = k 0 n k (l k a + l k l + 1) (a + ν l (k )) ν l ( GL el k(q) ) = l k a + l k l + 1 ν l ( GL el k+1(q) ) = lν l ( GL el k(q) ) + 1 ν l ( S k+1 ) = lν l ( S k ) + 1 = lν l ( GL el k(q) ) + 1 = ν l ( GL el k+1(q) ) S k+1 GL el k+1(q) Sylow l- (3) ν l ( GL n (q) ) = k 0 n k ν l ( GL el k(q) ) 1.5 Young Weyl + = + Weyl w γ(w) = w +

23 20 1 w S n w w(1) w(n) γ(w) = {(i, j) 1 i < j n, w(i) > w(j)} 1.12 s i S n i i + 1 (i, i + 1) s i (i) = i + 1, s i (i + 1) = i, s i (k) = k (k i, i + 1) {s i } 1 i<n S n Coxeter 1.13 w S n Coxeter w = s i1 s ir w r w w l(w) 1.9 w, s i w S n γ(w) + 1 (w 1 (i) < w 1 (i + 1)) γ(s i w) = γ(w) 1 (w 1 (i) > w 1 (i + 1)) {(j, k) j < k, s i w(j) > s i w(k)} (i) j, k {w 1 (i), w 1 (i + 1)} (ii) j = w 1 (i), k w 1 (i + 1) (iii) j = w 1 (i + 1), k w 1 (i) (iv) j w 1 (i + 1), k = w 1 (i) (v) j w 1 (i), k = w 1 (i + 1) (vi) {j, k} = {w 1 (i), w 1 (i + 1)} (i) (v) s i w(j) > s i w(k) w(j) > w(k)

24 1.5 Young w S n w = s i1 s ir γ(w) < r j < k w = s i1 ŝ ij ŝ ik s ir 1.9 γ(s i w) = γ(w) + 1 w 1 (i) < w 1 (i + 1) γ(s i w) = γ(w) 1 w 1 (i) > w 1 (i + 1) γ(w) = γ(w 1 ) γ(ws i ) = γ(w) + 1 w(i) < w(i + 1) γ(ws i ) = γ(w) 1 w(i) > w(i + 1) 1 k < r s i1 s ik (i k+1 ) < s i1 s ik (i k+1 + 1) 1 k < r γ(s i1 s ik+1 ) = γ(s i1 s ik ) + 1 γ(w) = r γ(w) < r 1 k < r s i1 s ik (i k+1 ) > s i1 s ik (i k+1 + 1) i k+1 < i k s ik, s ik 1,, s i1 j a = s ij+1 s ik (i k+1 ), b = s ij+1 s ik (i k+1 + 1) a < b s ij (a) > s ij (b) (1) a, b {i j, i j + 1} (2) a = i j, b {i j, i j + 1} (3) a = i j + 1, b {i j, i j + 1}

25 22 1 (4) a {i j, i j + 1}, b = i j (5) a {i j, i j + 1}, b = i j + 1 a = i j, b = i j + 1 w(i) = k, w(i + 1) = k ws i w 1 = (k, k ) s ij+1 s ik (i k+1 ) = i j, s ij+1 s ik (i k+1 + 1) = i j + 1 s ij+1 s ik s ik+1 s ik s ij+1 = s ij s ij s ij+1 s ik = s ij+1 s ik+1 w = s i1 s ij s ij+1 s ik+1 }{{} s ir s ij s ik w s ij s ik w S n γ(w) = l(w) w = s i1 s ir 1.9 γ(w) γ(s i2 s ir ) + 1 r = l(w) γ(w) < r s i1 s ir 1.14 S n Coxeter S = {s i } 1 i<n I S S n I S n Young S I 12 n s i I i i + 1 µ = (µ 1, µ 2, ) I µ I µ S I µ 1.15 µ = (µ 1, µ 2, ) I S µ n µ n

26 1.5 Young 23 I µ S Young S µ S µ = S µ1 S µ µ, ν n µ ν I µ I ν µ ν S ν = S ν1 S ν2 Young S µ 1.11 w = s i1 s ir w S µ {s i1,, s ir } I µ w S µ γ(sw) > γ(w) s I µ γ(w) = 0 w w S n γ(s i w) > γ(w) (1 i < n) 1.9 w 1 (1) < w 1 (2) < < w 1 (n) w 1 (i) = i (1 i n) γ(w) γ(w) = 0, 1 γ(w) > 1 γ(sw) < γ(w) s I µ sw = ss i1 s ir w = s i1 s ir S νi s s ik l(sw) = γ(sw) = γ(w) 1 = l(w) 1 s i1 ŝ ik s ir = sw S µ s ik s i1,, s ir I µ s ik S µ s ik I µ 1.12 µ n, ν n w S n S µ ws ν d µν

27 24 1 l(v 1 d µν v 2 ) = l(v 1 ) + l(d µν ) + l(v 2 ) v 1 S µ, v 2 S ν v 1 d µν v 2 u S µ ws ν u S µ ws ν v 1 S µ v 2 S ν u = v 1 uv 2 v 1, v 2 l(u ) = l(v 1 ) + l(u) + l(v 2 ) u, v 1, v 2 u = s i1 s ir, v 1 = s j1 s jm, v 2 = s k1 s kl u = s j1 s jm s i1 s ir s k1 s kl l(u ) < m + r + l v 1 s j1 s jm v 2 s k1 s kl u s i1 s ir s j s i u S µ ws ν s i s k u S µ ws ν l(u) s j s k v 1 = s j1 ŝ j s jm, v 2 = s k1 ŝ k s kl u = v 1uv 2 l(u ) = l(v 1 ) + l(u) + l(v 2 )

28 1.5 Young 25 u l(u ) = l(v 1 ) + l(u) + l(v 2 ) v 1 S µ, v 2 S ν u = v 1 uv 2 l(u ) = l(u) l(v 1 ) = l(v 2 ) = 0 v 1, v 2 u = v 1 uv 2 = u 1.13 w S n S µ ws ν u = d µν S µ ws ν l(su) > l(u) ( s I µ ) l(us) > l(u) ( s I ν ) S µ ws ν u d µν u l(su) l(u) l(us) l(u) u 1.12 v 1 S µ, v 2 S ν u = v 1 d µν v 2 l(u) = l(v 1 ) + l(d µν ) + l(v 2 ) v 1 s I µ l(sv 1 ) < l(v 1 ) 1.9 l(su) l(sv 1 ) + l(d µν ) + l(v 2 ) < l(u) v 2 v 1, v 2 u = d µν. Young 1.17 S µ \S n /S ν D µν = { w S n l(su) > l(u) ( s I µ ), l(us) > l(u) ( s I ν ) } D µν distinguished

29 26 1 w S n w(1) w(n) s i w(1) w(n) i i + 1 s i i i + 1 a i = i 1 k=1 µ k + 1, b i = i 1 k=1 ν k + 1 D µν 1, 2,, n w(1) w(n) a i,, a i+1 1 (i = 1, 2, ) w(b i ) < w(b i + 1) < < w(b i+1 1) (i = 1, 2, ) 1.3 n = 4, µ = (2, 2) 4, ν = (1, 2, 1) 4 I µ = {s 1, s 3 }, I ν = {s 2 } D µν = {1234, 1342, 3124, 3142} 1342 = s 2 s 3, 3124 = s 2 s 1, 3142 = s 2 s 1 s s 2 s 1 s

30 1.5 Young ν n S ν = {w S n l(ws) > l(w) ( s I ν )} u S ν, v S ν l(uv) = l(u) + l(v) µ = (1 n ) S ν = D µν 1.12, 1.13 u us ν u v u = s i1 s ir, v = s j1 s jt l(uv) < r + t uv = s i1 s ir s j1 s jm u, v uv = s i1 ŝ i s ir s j1 ŝ j s jt u us ν l(uv) = l(u) + l(v) 1.15 s, t Coxeter, w S n l(sw) = l(wt) l(swt) = l(w) sw = wt l(wt) > l(w) l(sw) > l(w) swt s i1 s ir l(w) = r w = ss i1 s ir t w = ss i1 ŝ i ŝ i s ir t swt = s i1 ŝ i ŝ i s ir swt w = ss i1 ŝ i s ir t sw = s i1 ŝ i s ir l(sw) r 1 l(sw) > l(w) = l(swt) = r w = ŝs i1 ŝ i s ir t wt = s i1 ŝ i s ir l(wt) r 1 l(wt) > l(w) = r w = s i1 s ir = swt sw = wt l(wt) < l(w) w = wt l(w t) > l(w )

31 28 1 (1) l(sw ) = l(swt) = l(w) = l(w t) (2) l(sw t) = l(sw) = l(wt) = l(w ) w = sw t wt = sw Deodhar 1.16 µ n µ S = {w S n l(sw) > l(w) ( s I µ )} Coxeter s v µ S (a) vs µ S. (b) Coxeter t I µ vs = tv vs µ S t I µ l(tvs) < l(vs) v µ S l(tv) > l(v) l(vs) > l(tvs) l(tv) 1 l(v) l(tv) = l(v) + 1 = l(vs) l(tvs) = l(v) 1.15 tv = vs 1.1 d D µν d 1 S µ d S ν = S d 1 I µd I ν S d 1 I µ d I ν d 1 S µ d S ν w d 1 S µ d S ν dw S µ d ds ν v S µ dw = vd v S µ, w S ν d D µν 1.14 l(d) + l(w) = l(dw) = l(vd) = l(v) + l(d) l(w) = l(v) w = s i1 s ir d, ds i1, ds i1 s i2,, ds i1 s ir d i µ S (0 i r) (1) d 0 = d

32 1.6 Tits 29 (2) t 1,, t k I µ {1} ds i1 s ik = t 1 t k d k Deodhar d k s ik+1 = t k+1 d k+1 (t k+1 I µ {1}, d k+1 µ S) (a) d k s ik+1 (b) d k s ik+1 µ S t k+1 = 1, d k+1 = d k s ik+1. µ S t k+1 I µ, d k+1 = d k. vd = dw = t 1 t r d r v, t 1,, t r S µ d, d r µ S d r = d, v = t 1 t r l(v) = l(w) = r t k = 1 d k = d (0 k r) = d 1 t k d d 1 I µ d I ν w S d 1 I µd I ν s ik 1.6 Tits Tits BN 1.18 G G (B, N) (a) G B N (b) T = B N N (c) W = N/T 2 {s i } i I (d) s i Bs i B. (e) w W s i Bw Bs i wb BwB. (B, N) Tits BN (d) (e) n i T = s i nt = w N n i n n i Bn i B n i Bn Bn i nb BnB n i n

33 G (B, N) BN BN (f) B W wbw 1 (w W ) w W wbw 1 = T. (g) C G (T ) = T Hopf T (E ) r E[T ] E[T 1, T 1 1,, T r, T 1 r ] (h) B U (i) B T U B = T U U T = 1. (ii) U G GL n (E) u U (u 1) n = 0 GL n (E) Tits Bruhat G = w W BwB BN Bruhat 1.2 S n GL n (E) GL n (q) (1) GL n (E) = B n wb n. w S n (2) GL n (q) = B n (q)wb n (q). w S n (2) B n (q) op

34 GL n (q) = B n (q) op wb n (q) w S n (e 1,, e n ) 1.6 Tits 31 g GL n (q) g (i, 1) 1 i < i 1 0 i = i λ , λ 1 g = (e i1 ) λ i 1 (1, 0,, 0) i 1 1 (e i1 ) (e i1, e i2 ) (e i1, e i2,, e in ) S n GL n (q) = w S n B n (q) op wb n (q) w 0 1 i n (i, n + 1 i) 1 0 w 0 S n B n (q) op = w0 1 B n(q)w 0 GL n (q) = w 0 GL n (q) = w S n B n (q)w 0 wb n (q)

35 32 1 B n (q)w 1 B n (q) = B n (q)w 2 B n (q) w 1 = g 1 w 2 h (g, h B n (q)) w 1 2 gw 1e w 1 1 (i) = w 1 2 ge i = w2 1 (g iie i + ) = g ii e w 1 2 (i) + he w 1 1 (i) e w 1 2 (i) w 1 2 (i) w 1 1 (i) w 1 1 g 1 w 2 e w 1 2 (i) = g 1 ii e w 1 1 (i) + h 1 e w 1 2 (i) e w 1 1 (i) w 1 1 (i) w 1 2 (i) w 1 = w Z[t] w S n t l(w) = (1 + t)(1 + t + t 2 ) (1 + t + + t n 1 ) B n (q)wb n (q)/b n (q) B n (q) wb n (q)/b n (q) B n (q) wb n (q)w 1 b = (b ij ) 1 i,j n B n (q) i > j b w(i)w(j) = 0 i < j b ij F q w 1 (i) < w 1 (j) B n (q)wb n (q)/b n (q) = GL n (q)/b n (q) = q n(n 1) 2 l(w 1) (q 1) n q n(n 1) 2 (q 1) n q n(n 1) 2 l(w 1) (q 1) n = ql(w) q n(n 1) 2 n (q i 1) i=1 = n 2 (q 1) n q n(n 1) i=1 q i 1 q 1

36 1.6 Tits 33 Bruhat t = q q p BN 1.3 s i B n (q)w B n (q)s i wb n (q) B n (q)wb n (q) (1) w 1 (i) < w 1 (i + 1) s i B n (q)w B n (q)s i wb n (q). (2) w 1 (i) > w 1 (i + 1) s i B n (q)w B n (q)wb n (q). w(k) = i k g B n (q)w k ge k = gw 1 e w(k) = gw 1 e ik gw 1 B n (q) i k i k 0 i k + 1,, n 0 ( ) g B n (q)w g k i (1) i k > i + 1 ( ) (2) i k = i + 1 =0 ) ( (3) i k = i =0 ) 0 ( (4) i k < i ( 0 0 ) i + 1 w(a) = i, w(b) = i + 1 a, b s i g s i B n (q)w k k a, b i k i, i + 1 (1) (4) i i + 1 s i g k i k 0 i k + 1,, n 0 k {a, b} i k = i i k = i + 1 (A) w 1 (i) < w 1 (i + 1) a < b a b i 0 = 0 i + 1 = 0

37 a 12 0, a 21 0 ( ) 0 a12 1 a 22 ( ) 0 a12 a 21 = a 21 a a 21 0 B n (q) a b i 0 = 0 i + 1 = 0 0 k {a, b} B n (q)s i w s i B n (q)w B n (q)s i wb n (q) (B) w 1 (i) > w 1 (i + 1) a > b b a i = 0 0 i + 1 = 0 ( b =0 ) Bn (q)s i wb n (q) ( ) b =0 0 =0 (1) 2 a 11 0, a 21 0, a 22 0 ( ) a a 22 a 21 = a 11 a 11a 22 a 21 a 21 a a 21 0 B n (q)

38 1.7 GL n (E) GL n (q) 35 b a i = 0 = 0 i + 1 = 0 0 B n (q)wb n (q) s i B n (q)w B n (q)wb n (q) (i) γ(s i w) = γ(w) + 1 s i B n (q)w B n (q)s i wb n (q). (ii) γ(s i w) = γ(w) 1 s i B n (q)w B n (q)wb n (q) s i B n w B n s i wb n B n wb n (1) w 1 (i) < w 1 (i + 1) s i B n w B n s i wb n. (2) w 1 (i) > w 1 (i + 1) s i B n w B n wb n. 1.7 GL n (E) GL n (q) GL n (E) GL n (q) GL n (q) 1.17 S µ Young B n (q)s µ B n (q) GL n (q). B n (q), S µ g B n (q)s µ B n (q) g 1 B n (q)s µ B n (q) w S µ w = s i1 s ir 1.11 s i1,, s ir I µ 1.3 s i1 B n (q)s i2 B n (q) s ir B n (q)s µ B n (q) B n (q)s µ B n (q) w S µ

39 36 1 B n (q)wb n (q)s µ B n (q) B n (q)s µ B n (q) B n (q)s µ B n (q) 1.20 S µ Young P µ (q) := B n (q)s µ B n (q) µ GL n (E) P µ := B n S µ B n µ F GL n (E) Frobenius q F (P µ ) = P µ P µ (q) = Pµ F Frobenius Weyl Coxeter P µ (q) P µ (q) = µ 1 µ GL n (q) µ n P µ (q) GL µ1 (q) GL µ2 (q) L µ (q) P µ (q) Levi P µ (q) L µ (q) 1 U µ (q) P µ (q) L µ (q) 1 U µ (q) P µ (q) P µ (q) U µ (q) 1.18 w S n w = s i1 s ir

40 1.7 GL n (E) GL n (q) 37 I(w) = {s S s = s ij j } µ n I µ = I(w) P µ (q) = B n (q), w l(w) l(w) = 0 l(w) > 0 w = s i2 s ir I µ = I(w ) P µ (q) = B n (q), w 1.4 s i1 B n (q)w B n (q)wb n (q) b 1, b 2, b 3 B n (q) s i1 b 1 w = b 2 wb 3 s i1 = b 2 wb 3 w 1 b 1 1 B n (q), w w = s i1 w B n (q), w P µ (q) = B n (q), w B n (q), w s i2,, s ir B n (q)s µ B n (q) = P µ (q) B n (q), w s i1 B n (q), w P µ (q) B n (q), s i1,, s ir B n (q), w P µ (q). P µ (q) = B n (q), w I(w) w supp(w) GL n (q) {P µ (q)} µ n 1.6 P GL n (q) µ n P = P µ (q) Bruhat S n W

41 38 1 P = w W B n (q)wb n (q) w W µ n I µ = w W I(w), I(w) = {s S s w }. W S µ P P µ (q) w W 1.18 I(w) B n (q)s I(w) B n (q) = B n (q), w P I P S µ P P µ (q) = B n (q)s µ B n (q) P P = P µ (q) 1.19 w S n (1) P µ (q)wp ν (q) = B n (q)s µ ws ν B n (q) (2) (P µ (q)wp ν (q)) S n = S µ ws ν 1.3 s i B n (q)w B n (q)s i wb n (q) B n (q)wb n (q) v S µ S µ B n (q)w B n (q)s µ wb n (q) wb n (q)s ν B n (q)ws ν B n (q) S µ wb n (q)s ν = w B n (q)s ν w S µw w S µw B n (q)w S ν B n (q) = B n (q)s µ ws ν B n (q) P µ (q)wp ν (q) = B n (q)(s µ B n (q)w)b n (q)s ν B n (q) B n (q)(s µ wb n (q)s ν )B n (q) B n (q)s µ ws ν B n (q) P µ (q)wp ν (q). P µ (q)wp ν (q) = B n (q)s µ ws ν B n (q) (1) Bruhat (B n (q)wb n (q)) S n = {w}

42 1.7 GL n (E) GL n (q) 39 (P µ (q)wp ν (q)) S n = (B n (q)s µ ws ν B n (q)) S n = S µ ws ν (2) 1.7 D µν P µ (q)\ GL n (q)/p ν (q) w P µ (q)wp ν (q) D µν P µ (q)\ GL n (q)/p ν (q) Bruhat (P µ (q), P ν (q))- P µ (q)wp ν 1.19(1) P µ (q)wp ν (q) = B n (q)s µ ws ν B n (q) w S µ \S n /S ν D µν P µ (q)w 1 P ν (q) = P µ (q)w 2 P ν (q) 1.19(2) S µ w 1 S ν = S µ w 2 S ν w 1, w 2 D µν w 1 = w {P µ (q)} µ n g GL n (q) P µ (q) = gp ν (q)g w D µν P µ (q) = wp ν (q)w 1 w(1) w(n) 1,, µ 1 w(i) = i (1 i µ 1 ) 1 a < b n w(a) > µ 1, w(b) µ 1 E ij 1 + F q E w(a)w(b) = w(1 + F q E ab )w 1 wp ν (q)w 1 = P µ (q) w P µ (q) = P ν (q) µ = ν P = wp µ (q)w 1 w S n Levi 1.22 P P = wp µ (q)w 1 w S n

43 40 1 U := wu µ (q)w 1 P L := wl µ (q)w 1 P Levi Levi L P = wp µ (q)w 1 L = {( 0 0 )} µ = (2, 2) 4 P = {( )} P = {( )}

44 2 Harish-Chandra 2.1 Harish-Chandra Lie Harish-Chandra Harish-Chandra Harish- Chandra F GL n (q)- Harish- Chandra G H G FG- M Ind G H(M) := FG FH M m 1 m (m M) ι : M Ind G H(M) (Ind G H(M), ι) FG- N FH- M Res G H(N) FG- Ind G H(M) N 1 F M N ι 1 Ind G H(M) N FG- mod f f M := f ι Hom FG (Ind G H(M), N) Hom FH (M, Res G H(N))

45 42 2 Harish-Chandra Frobenius f : M N FH- f Ind G H(f) : Ind G H(M) Ind G H(N) 1 Ind G H(f)(g m) = g f(m) (g G, m M) Ind G H : FH- mod FG- mod 2.1 (1) Frobenius FH- M FG- N 2 FH- M M FG- N N Hom FG (Ind G H(M), N) Hom FH (M, Res G H(N)) Hom FG (Ind G H(M ), N ) HomFH (M, Res G H(N )) (2) Ind G H ResG H (1) ψ : M Hom FG (Ind G H(M), N) M ϕ : N N f (ϕ f Ind G H(ψ)) M = ϕ f M ψ (2) Frobenius 2 Frobenius

46 2.1 Harish-Chandra Ind G H ResG H. G/H = {g i H} 1 i G/H FG (FH, FG)- FH- N Φ : Hom FH (FG, N) FG FH N = Ind G H(N) Φ(f) = g i f(g 1 i ) Φ FG- g G g 1 g i = g i h i (h i H) Φ(gf) = g i f(g 1 i = i g i h 1 i g) = g i f(h 1 i g 1 i ) f(g 1 i ) = gg i f(g 1 i ) = gφ(f) i Φ FG- Φ(f) = 0 f(g 1 ) = 0 f(hg 1 ) = hf(g 1 ) = 0 i G = Hg 1 i f G 0 Φ gi n i FG FH N f(hg 1 ) = hn i (h H) f Hom FH (FG, N) Φ(f) = g i n i Φ i i Φ FG- Φ N FH- N N Hom FH (FG, N) Ind G H(N) Hom FH (FG, N ) Ind G H(N ) FG- M M i Hom FG (M, Ind G H(N)) Hom FG (M, Hom FH (FG, N)) Hom FG (M, Ind G H(N )) Hom FG (M, Hom FH (FG, N )) Hom FG (M, Hom FH (FG, N)) Hom FH (FG FG M, N) = Hom FH (Res G H(M), N)

47 44 2 Harish-Chandra f Hom FG (M, Hom FH (FG, N)) g m f(m)(g) Hom FG (M, Hom FH (FG, N)) Hom FH (Res G H(M), N) Hom FG (M, Hom FH (FG, N )) HomFH (Res G H(M ), N ) 2.1 µ n, ν n µ ν P ν µ (q) := P µ (q) L ν (q), U ν µ(q) := U µ (q) L ν (q) e ν µ FUµ(q) ν e ν µ := 1 Uµ(q) ν u u Uµ ν (q) ν = (n) e ν µ e µ 1 Uµ(q) ν Pµ ν (q) L µ (q) 1 M FL µ (q)- Uµ(q) ν 1 M FP µ ν (q)- Infl ν µ(m) Infl P ν µ L µ (M). M Infl ν µ(m) Infl ν µ : FL µ (q)- mod FP ν µ (q)- mod {le ν µ l L µ (q)} FP µ ν (q)e ν µ leν µ = e ν µl (l L µ (q)) FPµ ν (q)e ν µ (FP µ ν (q), FL µ (q))- Infl ν µ(m) = FP ν µ (q)e ν µ FLµ(q) M 2.2 µ n, ν n µ ν R L ν L µ (M) := Ind Lν(q) P ν µ (q) Inflν µ(m)

48 2.1 Harish-Chandra 45 R Lν : FL µ (q)- mod FL ν (q)- mod L µ (q) L µ L ν (q) Harish-Chandra Pµ ν (q) R Lν M e ν µm Infl ν µ : FP ν µ (q)- mod FL µ (q)- mod L µ P ν µ l L µ (q) le ν µl 1 = e ν µ eν µm L µ (q)- f : M M FP µ ν (q)- Maschke M FUµ(q)- ν Infl ν µ : M = e ν µm (1 e ν µ)m e ν µm Res P ν µ (q) Uµ ν (q)(m) Infl ν µ(f) : e ν µm e ν µm f e ν µ M Infl ν µ 2.3 µ n, ν n µ ν R L ν L µ R L ν L µ (N) := Infl ν µ Res L ν(q) P ν µ (q)(n) : FL ν (q)- mod FL µ (q)- mod L ν (q) L µ (q) Harish-Chandra R L ν L µ Pµ ν Pµ ν (q) 2.1 R L ν L µ R L ν L µ. Res L ν(q) Pµ ν (q) IndL ν(q) (q) Infl ν µ Inflν µ P ν µ M FPµ ν (q)- mod, N FL µ (q)- mod M Infl ν µ(n) FU µ(q)- ν

49 46 2 Harish-Chandra f : M Infl ν µ(n) f : Infl ν µ(n) M FL µ (q)- e ν µm f M Infl ν µ(n), Infl ν f µ(n) M. f f e ν µm (1 e ν µ)m e ν µm e ν µm Hom FP ν µ (q)(m, Infl ν µ(n)) Hom FLµ (q)(e ν µm, N) f f Hom FP ν µ (q)(infl ν µ(n), M) Hom FLµ (q)(n, e ν µm) f f M M, N N Hom FP ν µ (q)(m, Infl ν µ(n)) Hom FLµ (q)(e ν µm, N) Hom FP ν µ (q)(m, Infl ν µ(n )) Hom FLµ (q)(e ν µm, N ) 2 M, N Infl ν µ Inflν µ 2.3 G := L (n) µ n, ν n µ ν R G L ν R L ν L µ (M) R G L µ (M), R L ν L µ R G L ν (N) R G L µ (N) U ν (q) U µ (q) e µ = ( 1 U ν µ(q) ) ( u u Uµ ν (q) 1 U ν (q) u U ν(q) u ) = e ν µe ν

50 2.1 Harish-Chandra 47 R L ν L µ RL G ν (N) = e ν µe (n) ν N = e (n) µ N = RL G µ (N) R G L µ (M) = F GL n (q) FPµ (q) FP µ (q)e µ FLµ (q) M = F GL n (q)e µ FLµ (q) M RL G ν R Lν L µ (M) = F GL n (q)e ν FLν (q) FL ν (q)e ν µ FLµ (q) M F GL n (q)e ν e ν µ FLµ (q) M = R G L µ (M) GL n (q) P ν (q) L ν (q) Pµ ν (q) = GL n (q) U ν (q) L µ (q) Uµ(q) ν = GL n(q) L µ (q) U µ (q) = GL n(q) P µ (q) R G L ν R Lν L µ (M) R G L µ (M) 2.4 G := L (n), µ n (1) P F GL n (q)- RL G µ (P ) FL µ (q)- (2) P FL µ (q)- RL G µ (P ) F GL n (q)- (1) P P F GL n (q) {P µ (q)g i } 1 i GLn (q)/p µ (q) P µ (q)\ GL n (q) FP µ (q)- F GL n (q) = FP µ (q)g i FP µ (q) GL n(q)/p µ (q) P F GL n (q) e µ R G L µ (P ) = e µ P (e µ FP µ (q)) GLn(q)/Pµ(q) FL µ (q)- e µ FP µ FL µ (q) RL G µ (P ) FL µ (q)-

51 48 2 Harish-Chandra (2) P P FL µ (q) Infl (n) µ (P ) = FP µ (q)e µ FLµ (q) P FP µ (q)e µ Infl (n) µ (P ) FP µ (q) RL G µ (P ) Ind GLn(q) P µ (q) (FP µ (q)) = F GL n (q) RL G µ (P ) F GL n (q)- 2.4 F GL n (q)- S cuspidal µ n FL µ (q)- X µ (n) Hom F GLn (q)(r G L µ (X), S) 0 S cuspidal cuspidal 2.5 S F GL n (q)- (1) S cuspidal (2) µ (n) RL G µ (S) = 0 G = L (n) (1) (2) µ (n) RL G µ (S) 0 FL µ (q)- X X Soc( RL G µ (S)) Hom F GLn (q)(r G L µ (X), S) Hom FLµ (q)(x, R G L µ (S)) 0. S cuspidal (2) (1) µ (n) RL G µ (S) = 0 FL µ (q)- X Hom F GLn (q)(r G L µ (X), S) Hom FLµ (q)(x, R G L µ (S)) = 0. S cuspidal

52 2.1 Harish-Chandra A A- M M Rad M M A Rad A A/ Rad A A- A/ Rad A- 2.6 A M A- P M A- p : P M P N p(n) M A- S P (S) P (S)/ Rad P (S) S A A = S P (S) dim S A- M S M [M : S] = dim Hom A (P (S), M) Hiss Dipper-Du 2.6 S F GL n (q)- P (S) S µ n G = L (n) triv (a) R G L µ (S) 0 (b) Hom F GLn (q)(ind GL n(q) U µ(q) (triv), S) 0 (c) FL µ (q)- Q Hom F GLn (q)(rl G µ (Q), S) 0

53 50 2 Harish-Chandra (d) R G L µ R G L µ (S) S (e) FL µ (q)- Q P (S) R L G µ (Q) (f) P (S) R G L µ R G L µ (P (S)) (a) (b) Ind P µ(q) U µ(q) Ind GL n(q) U µ(q) (triv) = Ind GL n(q) P µ(q) Ind P µ(q) U (triv) µ(q) (triv) FP µ(q)/u µ (q) FL µ (q)- FL µ (q) U µ (q) 1 Ind Pµ(q) U µ (q)(triv) Infl(n) µ (FL µ (q)) (triv) RL G µ (FL µ (q)) (a) Ind GLn(q) U µ (q) Hom F GLn(q)(Ind GLn(q) U µ (q) (triv), S) Hom F GLn(q)(R G L µ (FL µ (q)), S) Hom FLµ (q)(fl µ (q), R G L µ (S)) 0. (b) (c) FL µ (q) FL µ (q) Q: FL µ (q)- dim(q/ Rad Q) Q (triv) RL G µ (FL µ (q)) (b) Ind GL n(q) U µ (q) Hom F GLn (q)(r G L µ ( Q dim(q/ Rad Q) ), S) 0. Q Hom F GLn(q)(RL G µ (Q), S) 0. (c) (a) Hom FLµ(q)(Q, R G L µ (S)) Hom F GLn(q)(R G L µ (Q), S) 0 RL G µ (S) 0 (a) (d) [R G L µ R G L µ (S) : S] = dim Hom F GLn(q)(P (S), R G L µ R G L µ (S))

54 2.1 Harish-Chandra 51 = dim Hom FLµ (q)( R G L µ (P (S)), R G L µ (S)) P (S) S e µ RL G µ (P (S)) RL G µ (S) (a) RL G µ (S) 0 0-map Hom FLµ(q)( RL G µ (P (S)), RL G µ (S)) 0. (d) (a) (d) Hom FLµ (q)( R G L µ (P (S)), R G L µ (S)) 0 RL G µ (S) 0 (c) (e) Hom F GLn(q)(RL G µ (Q), S) R G L µ (Q) P (S) S 0 φ 0 RL G µ (Q). φ Rad P (S) P (S) Im(φ) Rad P (S) φ P (S) S 0-map φ RL G µ (Q) P (S) 0 P (S) P (S) RL G µ (Q) (e) (c) R L G µ (Q) P (S) P (S) S 0-map Hom F GLn (q)(rl G µ (Q), S) 0 (d) (f) (d) Hom F GLn(q)(P (S), RL G µ RL G µ (S)) 0 Hom F GLn (q)(r G L µ R G L µ (P (S)), S) 0

55 52 2 Harish-Chandra P (S) S 0 0 RL G µ RL G µ (P (S)) RL G µ RL G µ (P (S)) P (S) 0 P (S). P (S) RL G µ RL G µ (P (S)) (f) (d) R L G µ RL G µ (P (S)) P (S) P (S) S Hom F GLn(q)(R G L µ R G L µ (P (S)), S) 0 Hom F GLn (q)(p (S), R G L µ R G L µ (S)) Hom FLµ (q)( R G L µ (P (S)), R G L µ (S)) Hom F GLn (q)(r G L µ R G L µ (P (S)), S) 0 [RL G µ RL G µ (S) : S] Mackey 2.7 µ, ν n, w D µν, P 1 = P µ, P 2 = wp ν w 1 Levi L 1 = L µ, L 2 = wl ν w 1 U 1 = U µ, U 2 = wu ν w 1 (1) µ µ P µ L 1 L 1 (2) µ µ P µ (q) L µ (q) L µ (q) (3) L 1 P 2 L 1 L 2 Levi L 1 (4) L µ (q) wp ν (q)w 1 L µ (q) wl ν (q)w 1 Levi L µ (q)

56 2.2 Mackey 53 (5) P 1 P 2 P 1 U 2 P 1 L 2 (6) P 1 (q) P 2 (q) P 1 (q) U 2 (q) P 1 (q) L 2 (q) (1) I µ I µ I µ L 1 Q := (B n L 1 )S µ (B n L 1 ) Q = P µ L 1 Q (B n S µ B n ) L µ = P µ L µ Q P µ L µ L µ Bruhat L µ = (B n L µ )S µ (B n L µ ) P µ L µ b 1, b 2 B n L µ, u S µ b 1 ub 2 b 1 ub 2 P µ = B n S µ B n u S µ P µ L µ (B n L µ )S µ (B n L µ ) = Q. (2) P µ (q) L µ (q) P µ L 1 Frobenius (1) (3) I µ = I µ wi ν w 1 µ n µ µ I 1 = {w(1),, w(ν 1 )}, I 2 = {w(ν 1 + 1),, w(ν 1 + ν 2 )}, I (1) = {1,, µ 1 }, I (2) = {µ 1 + 1,, µ 1 + µ 2 }, = I a I (α) w(1)w(2) w(n) 1,, µ 1 I a (α) I (1) 1 = {1,, I (1) 1 }, I(1) 2 = { I (1) 1 + 1,, I(1) 1 + I(1) 2 }, I (α) = I (α) 1 I (α) 2 s i I µ {i, i + 1} a,α I (α) a

57 54 2 Harish-Chandra µ = ( I (1) 1, I(1) 2,, I(2) 1, I(2) 2,, ) P µ, L µ, P 1, L 1, P 2, L 2 { ( ) } i I (α), j I (β) (α > β) P µ = g = (g ij ) G i I a (α), j I (α) g ij = 0 b (a > b) { ( ) } i I (α), j I (β) (α β) L µ = g = (g ij ) G i I a (α), j I (α) g ij = 0 b (a b) { ( ) } P 1 = g = (g ij ) G i I (α), j I (β) (α > β) g ij = 0 { ( ) } L 1 = g = (g ij ) G i I (α), j I (β) (α β) g ij = 0 { ( ) } P 2 = g = (g ij ) G i I a, j I b (a > b) g ij = 0 { ( ) } L 2 = g = (g ij ) G i I a, j I b (a b) g ij = 0 L µ = L 1 L 2 L 1 P 2 = P µ L 1 (1) P µ L 1 = (B n L 1 )S µ (B n L 1 ) L 1 P 2 L µ (= L 1 L 2 ) Levi L 1 1 U µ L µ P µ L µ L µ 1 L 1 P 2 L 1 L 2 L 1 P 2 U µ L µ = L 1 U 2 (4) Frobenius (3) (5) U 1 P 2 P 1 P 2 L 1 U 1 = 1 P 1 P 2 = (L 1 P 2 )(U 1 P 2 ) p P 1 p = lu (l L 1, u U 1 ) l p p P 1 P 2 l L 1 P 2

58 2.2 Mackey 55 u U 1 P 2 (6) Frobenius (5) 2.7(4) R Lµ L µ wl νw 1 L µ wp νw : F(L 1 µ (q) wl ν (q)w 1 )- mod FL µ (q)- mod w 1 D νµ w 1 P µ (q)w L ν (q) w 1 L µ (q)w L ν (q) Levi L ν (q) R wl νw 1 L µ wl νw 1 P µ wl νw : F(wL 1 ν (q)w 1 )- mod F(L µ (q) wl ν (q)w 1 )- mod 2.7 µ, ν n FL ν (q)- M g wl ν (q)w 1 g m := (w 1 gw)m FwL ν (q)w 1 - w M w M G, H, K G Res G K Ind G H : FH- mod FK- mod Ind K L Res H L (L K H) Mackey Harish-Chandra Mackey 2.1 µ, ν n, G = L (n). R L µ L µ wl ν w 1 := R L µ R wl νw 1 L µ wl νw 1 L µ wl ν w 1 L µ wp ν w 1 := R wl νw 1 L µ wl νw 1 P µ wl νw 1 FL ν (q)- M R G L µ R G L ν (M) w D µν R Lµ L µ wl ν w 1 R wl νw 1 L µ wl ν w 1 ( w M)

59 56 2 Harish-Chandra M Res GL n(q) P µ(q) M(w) = FP µ (q)wp ν (q) Res GLn(q) P µ (q) Ind GL n(q) P ν(q) (Infl P ν L ν (M)) 1.7 GL n (q) = P µ (q)wp ν (q) w D µν FP ν (q) Ind GLn(q) P ν (q) (Infl Pν L ν (M)) FL µ (q)- Infl P ν L ν (M) w D µν M(w) e µ M(w) R Lµ L µ wl νw 1 L µ wp νw 1 R wl νw 1 L µ wl νw 1 P µ wl νw 1 ( w M) e µ = 1 U µ (q) u U µ (q) w m w m (m M) Res wp ν(q)w 1 P µ (q) wp ν (q)w ( w Infl P 1 ν L ν (M)) wp ν (q) u FP ν(q) Infl P ν L ν (M) M(w) FP µ (q) wp ν (q)w 1 - FP µ (q)- Ind Pµ(q) P µ(q) wp ν(q)w 1 Res wp ν(q)w 1 P µ(q) wp ν(q)w 1 ( w Infl Pν L ν (M)) M(w) P µ (q)wp ν (q)/p ν (q) P µ (q) wp ν (q)/p ν (q) P µ (q) wp ν (q)w 1 M(w) P µ (q) P µ (q) wp ν (q)w 1 dim M

60 2.2 Mackey 57 e µ M(w) e µ Ind P µ(q) P µ(q) wp ν(q)w 1 Res wp ν(q)w 1 P µ(q) wp ν(q)w 1 ( w Infl P ν L ν (M)) w Infl Pν L ν (M) wu ν (q)w 1 Res wp ν(q)w 1 P µ (q) wp ν (q)w 1 ( w Infl P ν L ν (M)) Infl P µ wp ν w 1 P µ wl ν w 1 Res wpν(q)w 1 P µ (q) wl ν (q)w 1 ( w Infl Pν L ν (M)) Infl Pµ wpνw 1 P µ wl νw 1 Res wl ν(q)w 1 P µ(q) wl ν(q)w 1 ( w M) N = Infl Pµ wpνw 1 P µ wl νw Res wl ν(q)w 1 1 P µ(q) wl ν(q)w ( w M) 1 e µ M(w) e µ Ind P µ(q) P µ (q) wp ν (q)w 1 (N) P µ (q) wp ν (q)w 1 U µ (q) wp ν (q)w 1 L µ (q) wp ν (q)w 1 e = 1 U µ (q) wp ν (q)w 1 u u U µ (q) wp ν (q)w 1 (FL µ (q), F(P µ (q) wp ν (q)w 1 ))- e µ FP µ (q) FL µ (q) e FP µ (q) wp ν (q)w 1 FL µ (q) wp ν (q)w 1 l e p (l L µ (q), p P µ (q) wp ν (q)w 1 ) e µ le p = e µ lp e µ M(w) e µ Ind P µ(q) P µ (q) wp ν (q)w 1 (N) Ind L µ(q) L µ (q) wp ν (q)w 1 (e N) L µ (q) wp ν (q)w 1 - e N X FP µ (q) wl ν (q)w 1 - P µ (q) wl ν (q)w 1 U µ (q) wl ν (q)w 1 L µ (q) wl ν (q)w 1

61 58 2 Harish-Chandra e = 1 U µ (q) wl ν (q)w 1 u u U µ (q) wl ν (q)w 1 e X FL µ (q) wl ν (q)w 1 - e Infl P µ wp ν w 1 P µ wl ν w 1 (X) Infl L µ wp ν w 1 L µ wl ν w 1 (e X) 2.7(6) U µ (q) wp ν (q)w 1 = (U µ (q) wl ν (q)w 1 )(U µ (q) wu ν (q)w 1 ) Infl P µ wp ν w 1 P µ wl ν w (X) U 1 µ (q) wu ν (q)w 1 e Infl Pµ wpνw 1 P µ wl νw (X) e X 1 L µ (q) wp ν (q)w 1 L µ (q) wu ν (q)w 1 Infl Lµ wpνw 1 L µ wl νw (e X) 1 X = Res wl ν(q)w 1 P µ (q) wl ν (q)w ( w M) N = Infl P µ wp ν w 1 1 P µ wl ν w (X) 1 e N Infl Lµ wpνw 1 L µ wl νw 1 (e Res wl ν(q)w 1 P µ(q) wl ν(q)w 1 ( w M)) e Res wl ν(q)w 1 P µ (q) wl ν (q)w 1 ( w M) = R wl νw 1 L µ wl ν w 1 P µ wl ν w 1 ( w M) e N Infl Lµ wpνw 1 L µ wl νw 1 ( R wl νw 1 L µ wl νw 1 P µ wl νw 1 ( w M)) e µ M(w) Ind L µ(q) L µ (q) wp ν (q)w (e N) 1 e µ M(w) R L µ L µ wl ν w 1 L µ wp ν w 1 R wlνw 1 L µ wl ν w 1 P µ wl ν w 1 ( w M) M

62 2.3 Howlett-Lehrer Howlett-Lehrer Harish-Chandra RL P G P F GL n (q)- RL G, RL G Dipper-Du Howlett-Lehrer 2.8 w a,b w a,b = ( ) 1 a a + 1 a + b 1 + b a + b 1 b n a + b 0 c n (a + b) S a+b S c S a+b S n (a+b+c) S n w a,b w a,b [c] l(w a,b [c]) = l(w a,b ) = ab S a+b S a S b ( ) ( ) 1 a + b 1 a a + 1 a + b w 0 =, w 0 = a + b 1 a 1 a + b a + 1 w a,b = w 0 w 0 Weyl w a,b w a,b Howlett-Lehrer Weyl 2.9 µ n µ = (µ 1,, µ r ) n σ S r µ = (µ σ(1),, µ σ(r) ) µ µ 2.8 µ µ w S n (i) wi µ w 1 = I µ (ii) s k I µ w(k) < w(k + 1)

63 60 2 Harish-Chandra µ = µ (0), µ (1),, µ (r) = µ w 1,, w r S n (a) w = w r w 1 w i w a,b [c] (b) l(w) = r l(w i ) (c) i=1 (i) w i I µ (i 1)w 1 i = I µ (i) (ii) s k I µ (i 1) w i (k) < w i (k + 1) l(w). l(w) = 0 I µ = I µ µ = µ. l(w) > 0 i w(i) > w(i + 1) s i I µ s i I µ µ 12 n 12 } {{ }}{{} µ 1 µ 2 i }{{} µ d i + 1 } {{ } µ d+1 d µ d µ d+1 µ (1) a = µ d, b = µ d+1, c = d 1 j=1 (1) w 1 I µ w1 1 = I µ (1) (2) s k I µ w 1 (k) < w 1 (k + 1) µ j w 1 = w a,b [c] w = ww1 1 (i) (1) w I µ (1)w 1 = Iµ s k I µ (1) (1) (2) (w1 1 (k), w 1 1 (k + 1)) I µ 1 (ii) w 1 (k) < w 1 1 (k + 1) w (k) = ww1 1 (k) < ww 1 1 (k + 1) = w (k + 1).

64 2.3 Howlett-Lehrer 61 l(w) = l(w ) + l(w 1 ) µ (2),, µ (r) w 2,, w r α = w(i), β = w(i + 1) k µ d µ d+1 i µ d + 1 k < k + 1 i i + 1 k < k + 1 i + µ d+1 ws k w 1 = (w(k), w(k + 1)) I µ w(k) < w(k + 1) w(k + 1) = w(k) + 1 α > β i µ d + 1,, i i + 1,, i + µ d+1 } {{ }} {{ } µ d µ d+1 w µ β, β + 1, β + µ d+1 1 } {{ } µ d+1 α µ d + 1,, α } {{ } µ d µ 1 n w 1 i µ d µ d+1 w 1 (i) = w 1 (i) γ(w 1 ) = γ(w 1 ) + µd µ d+1 l(w) = l(w ) + l(w 1 ) 2.2 µ µ w S n (i) wi µ w 1 = I µ (ii) s k I µ w(k) < w(k + 1) e µ F GL n (q)e µ we µ I µ

65 62 2 Harish-Chandra I µ = 0 µ = µ = (1 n ) e µ F GL n (q)e µ we µ l(w) l(w) = 0 l(w) = 1 s S w = s e µ se µ se µ F GL n (q)e µ se µ q (n 2) (= Uµ (q) ) u U µ(q) e µ suse µ F GL n (q)e µ se µ U 1 = {u U µ (q) sus B n (q)} U 2 = {u U µ (q) sus B n (q)sb n (q)} sb n (q)s B n (q) B n (q)sb n (q) U µ (q) = U 1 U 2 u U 1 sus = sus 1 B n (q) µ = (1 n ) U µ (q) = U n (q) sus U µ (q) e µ suse µ = e µ U 1 = U µ (q) s 1 U µ (q)s = U 1 = q (n2) 1 u U 1 e µ suse µ = q (n 2) 1 e µ u U 2 t T n (q), u, u U µ (q) sus = tu su te µ = e µ t e µ suse µ = t(e µ u )s(u e µ ) = te µ se µ F GL n (q)e µ se µ

66 2.3 Howlett-Lehrer 63 u U 2 e µ suse µ F GL n (q)e µ se µ q (n2) 1 e µ F GL n (q)e µ se µ l(w) > 1 w = w s, l(w) = l(w ) + 1 e µ F GL n (q)e µ w e µ se µ l(w) = 1 e µ F GL n (q)e µ se µ F GL n (q)e µ w e µ se µ l(w s) = l(w ) + 1 w B n (q)s B n (q)wb n (q) U µ (q) = {u 1, u 2, } t i T n (q), u i, u i U µ (q) (i = 1, 2, ) w u i s = t i u iwu i 2) e µ w e µ se µ = q (n 2) q(n t i e µ we µ F GL n (q)e µ we µ. i=1 e µ F GL n (q)e µ w e µ se µ F GL n (q)e µ we µ I µ > 0 I ν I µ wi ν w 1 I µ µ ν I ν e ν F GL n (q)e ν we ν U ν (q) U µ ν (q) = L µ (q) U ν (q) U µ (q) ( e ν = e µ 1 ν e µ = U ν µ (q) u U µ ν (q) s k I µ (ii) w(k) < w(k + 1) u ) w(l µ (q) U n (q))w 1 U n (q) i < j u wu ν µ (q)w 1 (i, j) i j ν e µ s i, s i+1,, s j 1 I ν w 1 (i) w 1 (j) ν u (i, j) w 1 uw (w 1 (i), w 1 (j)) 0

67 64 2 Harish-Chandra wu µ ν (q)w 1 U ν (q) u U ν µ (q) e ν wuw 1 = e ν ( ) 1 e ν we ν = e ν w U ν µ u e µ (q) u U ν µ (q) ( ) 1 = e ν U ν µ wuw 1 we µ = e ν we µ (q) u U µ ν (q) e ν F GL n (q)e ν we ν e ν = e ν e µ e ν F GL n (q)e ν we µ = F GL n (q)e ν e µ we µ F GL n (q)e µ we µ l L µ (q) wlw 1 L µ (q) e µ e ν l F GL n (q)e µ (wlw 1 )we µ = F GL n (q)e µ we µ l(w) e µ F GL n (q)e µ we µ 2.8 w 1,, w r w = w r w 1 r. r = 0 µ = µ w r = 1 w = w a,b [c] e µ w 1 e µ we µ F GL n (q)e µ we µ u U µ (q) 1 n e µ w 1 uwe µ F GL n (q)e µ we µ c {}}{ µ : }{{} }{{} µ 1 µ k 1 µ : }{{} }{{} µ 1 µ k 1 }{{} µ k }{{} µ k+1 }{{} µ k+1 }{{} µ k }{{} }{{} w µ µ k µ k+1 µ µ k µ k+1 µ n

68 2.3 Howlett-Lehrer 65 µ = (µ 1,, µ k 1, µ k + µ k+1, µ k+2, ) w S eµ u U µ (q)( B n (q)) w 1 uw P eµ (q) 1.7 D µµ P µ (q)\ GL n (q)/p µ (q) D µµ S eµ P µ (q)\p eµ (q)/p µ (q) t D µµ S eµ U t = {u U µ (q) w 1 uw P µ (q)tp µ (q)} U µ (q) = U t t D µµ S eµ t D µµ S eµ e µ ( u U t w 1 uw)e µ F GL n (q)e µ we µ t D µµ t D µµ I ν = t 1 I µ t I µ ν = n 1.1 t 1 S µ t S µ = S ν ( 1 e ν = L µ (q) U ν (q) u L µ (q) U ν (q) u ) e µ e µ te ν = 1 L µ (q) U ν (q) u L µ (q) U ν (q) e µ (tut 1 )te µ. u L µ (q) U ν (q) tut 1 U µ (q) t D µµ s k I µ t(k) < t(k + 1) t(l µ (q) U n (q))t 1 U n (q) ts k t 1 I µ s k I ν tut 1 U µ (q) e µ te ν = e µ te µ u U t w 1 uw = u l tl u (u, u U µ (q), l, l L µ (q)) e µ (w 1 uw)e µ = l e µ te µ l = l e µ te ν l I ν I µ e ν l F GL n (q)e µ we µ

69 66 2 Harish-Chandra l e µ te ν l F GL n (q)e µ we µ ( e µ t D µµ S eµ ti µ t 1 =I µ u U t e µ (w 1 uw)e µ F GL n (q)e µ we µ u U t w 1 uw ) e µ F GL n (q)e µ we µ w(a, b ) (0 a a, 0 b b, a + b = a) 1, 2, a, a + 1,, a + b, a + 1,, a, a + b + 1,, a + b S a+b D µµ S eµ = {w(a, b )[c] 0 a a, 0 b b, a + b = a} a = a w(a, b )[c] = 1 t D µµ S eµ ti µ t 1 = I µ t 1 t = w(a, b )[c] 1 a < a (a, a + 1) I µ a = 0 a = b < b (a, a + b + 1) I µ a = b t = w, µ = µ ti µ t 1 = I µ t 1 t = w, µ = µ u U t=w w 1 uw = u l wl u (u, u U µ (q), l, l L µ (q)) e µ (w 1 uw)e µ = l e µ (wl w 1 )we µ = l e µ (wl w 1 )we µ = l (wl w 1 )e µ we µ e µ (w 1 uw)e µ F GL n (q)e µ we µ e µ ( u U 1 w 1 uw)e µ F GL n (q)e µ we µ

70 2.3 Howlett-Lehrer 67 U 1 = {u U µ (q) w 1 uw P µ (q)} = U µ (q) wp µ (q)w U µ (q) = I b., wp µ (q)w 1 = 0. 0 I a I a, I b a, b... U 1 = U µ (q) wp µ (q)w 1 = I b 0 0 I a U 1 wu µ (q)w 1 = I b 0 I a... u U 1 e µ (w 1 uw)e µ = e µ u U 1 e µ (w 1 uw)e µ = U 1 e µ e µ F GL n (q)e µ we µ r = 1 r > 1 w = w r w 2 w = w w 1 e µ (1) F GL n (q)e µ w e µ (1) w 1 e µ e µ F GL n (q)e µ (1)w 1 e µ F GL n (q)e µ w e µ (1)w 1 e µ w 1 = w a,b [c] µ µ (1) a b l(w) = l(w ) + l(w 1 ) µ (1) µ

71 68 2 Harish-Chandra a b I b w 1 Uµ (q)w U µ (1)(q) 0 I a... U µ (1)(q) w 1 U µ (q)w 1 1 =... I b 0 0 I a... r = 1 u U µ (1)(q) u U µ (q), u U µ (q) u = (w 1 u w )(w 1 u w1 1 ) = w 1 u wu w1 1 e µ w uw 1 e µ = e µ we µ u U µ (1) (q) e µ w uw 1 e µ = U µ (1)(q) e µ we µ e µ F GL n (q)e µ w e µ (1)w 1 e µ = F GL n (q)e µ we µ 2.2 µ, µ n, w S n P = P µ P = wp µ w 1 L = L µ Levi RL P G RL P G R G L P (M) = F GL n (q)e µ FL µ(q) RL P G (M) = F GL n(q)we µ w 1 M, FL µ(q) M

72 2.3 Howlett-Lehrer 69 L µ (q) = wl µ (q)w 1 w uwv (u S µ, v S µ ) (i) w 1 I µ w = I µ (ii) s k I µ w 1 (k) < w 1 (k + 1) 2.2 e µ F GL n (q)e µ w 1 e µ F GL n (q) g g 1 e µ e µ we µ F GL n (q) (i) (ii) (iii) wi µ w 1 = I µ (iv) s k I µ w(k) < w(k + 1) e µ F GL n (q)e µ we µ = F GL n (q)e µ e µ we µ ξ F GL n (q)e µ e µ = ξe µ we µ Φ : F GL n (q)e µ F GL n (q)e µ w 1, x xe µ we µ w 1 Φ(ξ) = ξe µ we µ w 1 = e µ w 1 Φ Φ x F GL n (q)e µ Φ(x) = 0 e µ e µ we µ F GL n (q) x = xe µ (xe µ we µ w 1 )wf GL n (q) = Φ(x)F GL n (q) = 0. Φ F GL n (q)- l L µ w 1 lw L µ (q) Φ(xl) = xle µ we µ w 1 = xe µ w(w 1 lw)e µ w 1 = xe µ we µ (w 1 lw)w 1 = Φ(x)l Φ (F GL n (q), FL µ (q))-.

73 70 2 Harish-Chandra 2.4 Harish-Chandra 2.10 F GL n (q)- S B(S) := {µ n R G L µ (S) 0} µ ν I µ I ν B(S) 2.9 S F GL n (q)- µ, ν B(S) w D µν wi ν w 1 = I µ P S 2.6 (a) (f) P R G L µ R G L µ (P ) R G L ν R G L ν (P ) 1 dim Hom F GLn (q)(p, S) dim Hom F GLn (q)(r G L µ R G L µ (P ), S) = dim Hom F GLn(q)(P, R G L µ R G L µ (S)) dim Hom F GLn(q)(R G L ν R G L ν (P ), R G L µ R G L µ (S)) = dim Hom F GLn (q)( R G L µ R G L ν R G L ν (P ), R G L µ (S)) RL G µ RL G ν ( RL G ν (P )) Mackey Q w = R wl νw 1 L µ wl ν w 1 ( w ( R G L ν (P ))) 2.4 Q w F(L µ (q) wl ν (q)w 1 )- dim Hom FLµ (q)(r Lµ L µ wl νw (Q 1 w ), RL G µ (S)) 0 w D µν w D µν F(L µ (q) wl ν (q)w 1 )- Q Hom FLµ (q)(r L µ L µ wl ν w 1 (Q), R G L µ (S))

74 2.4 Harish-Chandra 71 Hom F GLn (q)(rl G µ wl νw 1(Q), S) 0 I µ = I µ wi ν w 1 2.7(3) L µ (q) wl ν (q)w 1 = L µ (q) R G L µ 2.6 (a) (c) (S) 0 µ I µ = I µ I µ wi ν w 1 µ ν w D νµ I ν w I µ w 1 I µ = I ν I µ = wi ν w S F GL n (q)- µ B(S) RL G µ (S) X 1, X 2 w S n (i) wl µ (q)w 1 = L µ (q) (ii) X 1 w X 2 FL µ (q)- X 1 Soc( RL G µ (S)) X 2 w S n (i) (ii) P (X 2 ) X 2 [ RL G µ (S) : X 2 ] 0 Hom F GLn(q)(R G L µ (P (X 2 )), S) Hom FLµ(q)(P (X 2 ), R G L µ (S)) 0 RL G µ (P (X 2 )) S S P (S) P (S) S 0 φ 0 RL G µ (P (X 2 )) φ Rad P (S) P (S) Im(φ) Rad P (S) φ P (S) S φ

75 72 2 Harish-Chandra RL G µ (P (X 2 )) P (S) 0 P (S) P (S) R L G µ (P (X 2 )) X 1 0 Hom FLµ (q)(x 1, R G L µ (S)) Hom F GLn (q)(r G L µ (X 1 ), S) RL G µ (X 1 ) S [RL G µ (X 1 ) : S] 0 Hom F GLn(q)(P (S), R G L µ (X 1 )) 0 P (S) R L G µ (P (X 2 )) Hom F GLn (q)(r G L µ (P (X 2 )), R G L µ (X 1 )) 0 Mackey 1 dim Hom F GLn (q)(r G L µ (P (X 2 )), R G L µ (X 1 )) = dim Hom FLµ (q)( RL G µ RL G µ (P (X 2 )), X 1 ) = dim Hom FLµ(q)(R L µ L µ wl µ w R wl µw 1 1 L µ wl µ w ( w P (X 1 2 )), X 1 ). w D µµ w D µµ F(L µ (q) wl µ (q)w 1 )- Q Q R wlµw 1 L µ wl µw ( w P (X 1 2 )) R L µ L µ wl µ w 1 (Q) X 1 RL G µ R G L µ wl µ w 1(Q) RG L µ (X 1 ) RL G µ (X 1 ) S RL G µ wl µw 1(Q) S

76 2.4 Harish-Chandra 73 I ν = I µ wi µ w (a) (c) RL G ν (S) 0 ν B(S) µ I ν = I µ I µ wi µ w 1 I µ = wi µ w 1 wl µ (q)w 1 = L µ (q) Q = R L µ L µ wl µ w 1 (Q) X 1 Q R wl µw 1 L µ wl µ w ( w P (X 1 2 )) = w P (X 2 ) Q = w P (X 2 ) w P (X 2 ) w X 2 X 1 w X µ = n X cuspidal FL µ (q)- (L µ, X) cuspidal 2.12 S F GL n (q)- cuspidal (L µ, X) RL G µ (X) S S (L µ, X) F GL n (q)- S (L µ, X)- µ B(S) ν < µ RL G ν (S) (a) (c) FL ν (q)- Q RL G ν (Q) S RL G µ (X) S 0 RL G ν (Q) Hom F GLn(q)(RL G ν (Q), RL G µ (X)) 0 Mackey R G L µ R G L ν (Q) w D µν R Lµ L µ wl ν w 1 R wlνw 1 L µ wl ν w 1 ( w Q) w D µν P = R wl νw 1 L µ wl νw ( w Q) 1

77 74 2 Harish-Chandra Hom FLµ (q)(r L µ L µ wl νw 1 (P ), X) 0 X cuspidal I µ wi ν w 1 = I µ I µ wi ν w 1 I µ I ν ν < µ I ν I µ µ B(S) 2.3 S F GL n (q)- cuspidal (L µ, X) S n S (L µ, X)- S (L µ, X)- µ B(S) X Soc( RL G µ (S)) ν < µ R L µ L ν (X) = 0 R L µ 0 X R G L µ (S) 0 R Lµ L ν (X) R Lµ L ν RL G µ (S) = RL G ν (S) µ RL G ν (S) = 0 R L µ L ν (X) = X cuspidal FL µ (q)- Hom F GLn (q)(r G L µ (X), S) Hom FLµ (q)(x, R G L µ (S)) 0 RL G µ (X) S S (L µ, X)- S (L µ, X)- (L ν, Y ) µ ν B(S) 2.9 w D µν wi ν w 1 = I µ wl ν (q)w 1 = L µ (q) 2.2 R G L µ ( w Y ) = R G L µ wp ν w 1(w Y ) F GL n (q)- RL G µ ( w Y ) RL G ν (Y ) m wm (m RL G ν (Y )) F GL n (q)- w RL G ν (Y ) RL G ν (Y ) RL G µ ( w Y ) w RL G ν (Y ) N = GL n (q) / P ν (q) GL n (q) P ν (q) L ν

78 2.4 Harish-Chandra 75 N GL n (q)/p ν (q) = h i P ν (q) i=1 w R G L ν (Y ) = N i=1 h i FP ν (q) Infl Pν(q) L ν (q) (Y ) wgw 1 (g GL n (q)) wgw 1 h i y = gh i y (y Y ) GL n (q)/wp ν (q)w 1 = RL G µ ( w Y ) = N wh i w 1 i=1 N (wh i w 1 )wp ν (q)w 1 i=1 1 Infl wp ν(q)w FwP ν(q)w 1 L µ(q) ( w Y ) wgw 1 (g GL n (q)) wgw 1 wh i w 1 y = w(gh i )w 1 y (y Y ) Ψ : w RL G ν (Y ) RL G µ ( w Y ) h i y wh i w 1 y gh i = h j p (p P ν (q)) Ψ(wgw 1 h i y) = Ψ(gh i y) = Ψ(h j p y) = Ψ(h j py) = wh j w 1 py = wh j w 1 (wpw 1 ) y = wh j pw 1 y = wgw 1 wh i w 1 y = wgw 1 Ψ(h i y) Ψ F GL n (q)- RL G ν (Y ) RL G µ ( w Y )

79 76 2 Harish-Chandra S (L ν, Y )- (L µ, w Y )- (L ν, Y ) (wl ν w 1, w Y ) = (L µ, w Y ) S (L µ, X)- (L µ, w Y )- µ = ν µ = ν Hom F GLn (q)(r G L µ (X), S) 0, Hom F GLn (q)(r G L µ (Y ), S) 0 X, Y Soc( RL G µ (S)) 2.11 µ B(S) 2.10 w S n (i) wl µ (q)w 1 = L µ (q) (ii) X w Y (L µ, X) (L µ, Y ) 2.3 { F GL n (q)- } = (L µ, X)- (L µ,x) µ X (1) {µ = n} µ ν w S n wi µ w 1 = I ν {µ n}/s n µ (2) µ {µ n}/s n cuspidal FL µ (q)- X Y w S n wl µ (q)w 1 = L µ (q) X w Y X

80 F GL n (q)- Harish-Chandra (1) cuspidal (2) Harish-Chandra A Hecke (3) A Hecke (4) l- F GL n (q)- (1) GL n (q) cuspidal Gelfand (2) modular Howlett-Lehrer GL n (q) Dipper (3) Specht (4) foot index head index G FG- G l- (1), (2), (3) F GL n (q)- foot index GL n (q) l- head index (4) F GL n (q)- GL 2 (q) GL 2 (q) ( ) ( ) α 0 α 1, 0 α 0 α (α F q ) ( α ) 0 0 β ({α β} F q )

81 78 2 Harish-Chandra ( ) 0 α q+1 1 α + α q ([α] = {α, α q } F q 2 \ F q ) α F q 2 X α (α F q ) f α = (X α)(x α q ) (α F q 2 \ F q ) µ : F P F α F q f α (1 2 ), (2) F q {α β} F q f α, f β (1) [α] F q 2 \ F q f α (1) 1 ϕ, ψ : F q F T 2 (q) 1 ϕ ψ ( ) α 0 ϕ(α)ψ(β) (α, β F q ) 0 β (T, ϕ ψ) cuspidal (T, ϕ ψ) Harish-Chandra GL 2 (q)- End F GL2 (q)(rt G (ϕ ψ))- 1 F q F q l- (i) ϕ ψ End F GL2 (q)(rt G (ϕ ψ)) F Harish- Chandra (ii) ϕ = ψ End F GL2(q)(R G T (ϕ ψ)) F[T ]/(T 2 (q 1)T q) Harish-Chandra (a) l 3 q 1 2 (b) l q + 1 1

82 (i) ( ) α 0 0 β ({α β} F q ) l- (ii) (b) (a) RT G (ϕ det) ϕ det 2 cuspidal F GL 2 (q)- (a) RT G (ϕ det) cuspidal F GL 2 (q)- 2.1 P 1 (F q ) 1 ( ) a b z = az + b c d cz + d (z P 1 (F q )) GL 2 (q) P 1 (F q ) F GL 2 (q)- L l q + 1 L M = { c x x c x F, cx = 0} N = F ( x) x P 1 (F q ) M/N cuspidal F GL 2 (q)- cuspidal (iii) ψ : F q F 1 U 2 (q)- Fv ( ) 1 b v = ψ(b)v 0 1 GL 2 (q) V 2 (q) {( ) } 1 b V 2 (q) = b F q, d F q 0 d

83 80 2 Harish-Chandra M = Ind V 2(q) (Fv) M FV 2(q)- U 2 (q) M FV 2 (q)- (a) l 3 q 1 F q \ F 2 q l- Frobenius (b) l q + 1 F q l- Frobenius 2 F GL 2 (q)- cuspidal F GL 2 (q)- (iii) (a) (b) F q 2 \ F q l- Frobenius ( ) 0 α q+1 1 α + α q ([α] = {α, α q } F q 2 \ F q ) l- (ii) (b) l q + 1 (iii) (b) F q l- cuspidal F GL 2(q)- ( ) α 0, 0 α ( ) α 1 0 α (α F q ) l- 2.1 cuspidal F q l- cuspidal F GL 2(q)-

84 3 Hall 3.1 GL n (q) Brauer GL n (q) Lie 3.1 s GL n (E) E n E n = λ E{v E n sv = λv} g GL n (E) s GL n (E) gsg 1 GL n (E) GL n (q) 3.2 u GL n (E) (u 1) n = 0 u GL n (E) gsg 1

85 82 3 Hall GL n (E) 3.1 g GL n (E) (i) g = su = us (ii) u s (s, u) (iii) s p u p g GL n (E) p r m m p x, y xp r + ym = 1 s = g xpr, u = g ym (i) s m = 1 s s (ii) (u 1) pr = u pr 1 = 0 u g = su = us (s, u) (s, u ) s, u g = s u s s = g xpr u u = g ym s 1 s = u u 1 1 s 1 s = u u 1 = g = su Jordan 3.4 G p (a) g G p- g p (b) g G p- g p p GL n (q) q p g GL n (E) Jordan g GL n (q) GL n (E) GL n (q) g GL n (q) p- 3.1

86 3.1 GL n (q) 83 GL n (q) F GL n (q)- l- l F 3.5 (a) g GL n (q) l- g l (b) g GL n (q) l- g l l = 0 l- l- l- l p g = su Jordan u l- s 3.2 s GL n (q) (i) s = s l s l = s l s l (ii) s l l- s l (s l, s l ) (iii) s l, s l 3.1 l- l- l- l- (iii) s p (s l, s l, u) g GL n (q) (s l, s l, u) (gs l g 1, gs l g 1, gug 1 ) 3.3 GL n (q) (s l, s l, u) l- s l = 1

87 84 3 Hall g = s l s l u g = s l s lu h GL n (q) g = hgh 1 s = s l s l, s = s l s l s u = (hsh 1 )(huh 1 ) hsh 1, huh s l s l = s = hsh 1 = (hs l h 1 )(hs l h 1 ), u = huh s l = hs l h 1, s l = hs l h 1 F q - 1 F q [X] F = {f(x) F q [X] f(0) 0, f(x) } f F f(x) = X d + a 1 X d a d (a 1,, a d F q ) J(f) J(f) = 0 a d a2 1 a 1 s GL n (q) X s F q [X]- s 1 F q [X] n = F n q F q F q [X] F q [X]- F q F n q 0 F q [X] n X s F q [X] n F n q 0 X X F q [X] X s m f Z 0

88 3.1 GL n (q) 85 F n q f F (F q [X]/(f)) m f f F q [X]/(f) F q deg(f) F q [X]- F n q s GL n(q) GL n (q) {(m f ) f F m f Z 0, f F m f deg(f) = n} 3.6 s GL n (q) C G (s) = {g GL n (q) gs = sg} s C G (s) p C G (s) 3.4 g = su GL n (q) (i) s GL n (q) (ii) u C G (s) GL n (q) s GL n (q) g C G (s) 3.3 s GL n (q) C G (s) C G (s) = Aut Fq[X](F n q ) := End Fq[X](F n q ) GL n (q) X s F n q F q[x]- F n q f F (F q [X]/(f)) m f (A) (B) (C)

89 86 3 Hall 3.5 (A) f F F q [X]/(f) F q [X]- (B) f, f F f f Hom Fq [X](F q [X]/(f), F q [X]/(f )) = 0. (C) f F End Fq [X](F q [X]/(f)) F q deg(f). F q [X]/(f) V 0 c V F q [X]/(f) c 1 f X 1 V V = F q [X]/(f) F q [X]/(f) F q [X]- (A) f, f F f f F q [X]- ϕ : F q [X]/(f) F q [X]/(f ) (A) ϕ F q [X]- f 0 f (f ) f (f) (f) = (f ) f, f f = f (B) ϕ End Fq[X](F q [X]/(f)) ϕ(1) = a F q [X]/(f) ϕ a ϕ ϕ(1) F q - End Fq [X](F q [X]/(f)) F q [X]/(f) (C) 3.7 λ = (λ 1, λ 2, ) n λ = n λ 1 λ 2 λ n P = {λ n n Z 0 }

90 3.1 GL n (q) 87 λ n λ Young Young k λ k 3.1 λ = (5, 2, 2, 1, 0, 0, ) λ Young λ n Young k t λ k λ t λ n t λ = ( t λ 1, t λ 2, ) t λ k > 0 t λ k = max{j λ j k} λ = (5, 2, 2, 1, 0, 0, ) t λ = (4, 3, 1, 1, 1, 0, ) Jordan GL n (q) n 3.6 s GL n (q) C G (s) = Aut Fq [X](F n q ) f F GL mf (F q deg(f)) (m f ) f F C G (s) µ(f) m f F P {µ : F P µ(f) m f }

91 88 3 Hall (B) (C) p Jordan X F q [X]/(f) F q deg(f) f(x) = 0 α f α f F q deg(f) s C G (s) 3.6 s s End Fq [X](F q [X]/(f)) X End Fq[X](F q [X]/(f)) F q [X]/(f) F q deg(f) 3.6 s C G (s) f F GL mf (F q deg(f)), s (α f ) f F α f m f s l, s l s s = s l s l F q deg(f) α f F q deg(f) l- l- 3.8 α F Frobenius {α qk k Z/dZ} [α] q d #[α] 3.9 I = ( d1,, d N m 1,, m N [α 1 ],, [α N ] µ (1),, µ (N) ) (n, l)-index (i) µ (i) m i. (ii) α i F q d i l- l- β i F q #[α d i iβ i ] = d i β i F q d i

92 3.1 GL n (q) 89 (iii) N m i d i = n. i=1 1,, N (n, l)-index l = 0 α i Frobenius [α] F q d i f [α] = x [α] (X x) F q [X] (n, 0)-index f F deg(f) µ(f) = n µ : F P 3.10 (n, l)-index ( d1,, d N m 1,, m N I = [α 1 ],, [α N ] µ (1),, µ (N) ) head (i) #[α i ] = d i. (ii) [α 1 ],, [α N ] 3.1 (1) GL n (q) (n, 0)-index (2) GL n (q) l- head (n, l)-index GL n (q) (n, 0)-index 3.6 g l- g = su Jordan s l- (m f ) f F C G (s) µ(f) m f µ : F P