CHARACTERS OF SL 2 (F q )

Size: px
Start display at page:

Download "CHARACTERS OF SL 2 (F q )"

Transcription

1 CHARACTERS OF SL 2 (F q ) 1. Recall and notations 1.1. Characters of finite groups Let Γ be a finite groups, K be a field of characteristic zero containing Γ -th roots of unity. We note by KΓ the group ring associated to the finite group Γ. As a result (Theorem 10.3 of [1]), the K-algebra KΓ is a splitting field for Γ, namely any irreducible K-representation of Γ is also K- irreducible. Moreover, any (finite dimensional) K-representation is afforded by a K-representation. Set Irr(Γ) the set of irreducible K-representations of Γ, then for any finite dimensional K-representation of Γ, we have the following decomposition V V (τ) eτ. τ Irr(Γ) Let (V, ρ) be a representation of Γ, recall that the character of this representation is the application defined by χ ρ : Γ K, γ Tr(ρ(γ): V V ). We set deg(χ ρ ) = dim K V, and the character χ ρ is called linear if deg(χ ρ ) = 1. The basic properties of the characters are 1. If ρ = τ σ, then χ ρ = χ τ + χ σ. 2. The character χ ρ depends only on the conjugate class: for any γ Γ and any g Γ, we have χ ρ (g 1 γg) = χ ρ (γ). 3. Two representations with the same character are isomorphic. c 1997, Société MathéMatique de France

2 2 CHARACTERS OF SL 2 (F q) 4. For two representations V V of Γ, we define their scalar product by < V, V > Γ = dim K Hom KΓ (V, V ). By Schur s lemma, if < V, V >= 1, then V V are two isomorphic irreducible representations of Γ. 5. The cardinal Irr(Γ) is equal to the number of conjugacy classes of Γ, and we have moreover χ(1) 2 = Γ Notations. χ Irr(Γ) Let Γ be a finite group, and KΓ be its group ring. Every representation considered in this report is of finite dimensional. We will denote by K 0 (KΓ) the Grothendieck groups of the category of finite dimensional Γ-representations. Which is then the same as the free abelian group generated by Irr(G). For V a KΓ-module, we denote by [V ] (or [V ] Γ when necessary) the element in the Grothendieck group K 0 (KΓ) corresponding to V In this report, we are interested in the characters of the group SL 2 (F q ). In the following, we will also use the following notations G = SL 2 (F q ), where q = p r for some odd prime p. U G the unipotent subgroup {( ) } 1 x U = : x F 0 1 q F q T G the multiplicative subgroup {( ) } a 0 T = 0 a 1 : a F q F q B G the solvable subgroup {( ) } a B = 0 a 1 : a F q. Recall also that in the group G = SL 2 (F q ), there are q + 4 conjugacy classes (Theorem of [2]). In particular, the number of characters is equal to q+4.

3 CHARACTERS OF SL 2 (F q) 3 2. Harish-Chandra Induction The main reference of this is the Chapter 3 of [2] Bimodules. Let Γ and Γ be two finite groups, and let M be a (KΓ, KΓ )-bimodule of finite type. In particular, the dual M = Hom K (M, K) is naturally a (KΓ, KΓ)-bimodule. Define the following two functors: and We will denote by F M : KΓ mod KΓ mod, V M KΓ V. F M : KΓ mod KΓ mod, V M KΓ V. F M : K 0 (KΓ ) K 0 (KΓ), and F M : K 0 (KΓ) K 0 (KΓ ) the two induced morphisms between the Grothendieck groups. Lemma 2.1. With the notations as before, we have and Hom KΓ (V, F M V ) Hom KΓ ( F M V, V ) Hom KΓ ( F M V, V ) Hom KΓ (V, F M V ). Proposition 2.2. For any χ K 0 (KΓ ) and any γ Γ, we have F M (χ )(γ) = 1 Γ Tr M (γ, γ )χ (γ 1 ) γ Γ A similar equality holds for F M. Proof. We consider M K V, and define the action of Γ op Γ on M K V by the following formula (γ 1, γ 2) m x = mγ 1 1 γ 2x. Since the element e = 1 Γ (γ 1, γ ) γ is an idempotent, we get the following decomposition (here we identify e with the endomorphism of M K V defined by e) (1) M K V = ker(e) Im(e). Now we claim that (2) ker(e) = Vect K { m x mγ 1 γ x m M, γ Γ, x V }

4 4 CHARACTERS OF SL 2 (F q) Indeed, by definition, e(m x mγ 1 γ x) = 0. On the other hand, let i m i x i M K V an element in ker(e), we find hence i ( ) e m i x i i m i x i = 1 Γ = 1 Γ m i γ 1 γ x i = 0 i,γ ( mi x i m i γ 1 γ ) x i. In this way, we get the equality (2). By consequence, i,γ Im(e) M KΓ V. Now, for any γ Γ, which will be identified with the automorphism of M K V defined by γ, the automorphism γ preserves the decomposition (1). We have then γ = γ e + γ (id e) Hence Tr M KΓ V (γ) = Tr Im(e)(e) = Tr M K V (γ e). Therefore F M (V )(γ) = Tr(γ e: M K V M K V ) = 1 Γ Tr M (γ, γ 1 ) Tr V (γ ). γ In terms of characters, we get for any χ K 0 (KΓ ). F M (χ )(γ) = 1 Γ Tr M(γ, γ 1 )χ (γ ) 2.2. Harish-Chandra induction. We consider the (KG, KT )-bimodule K[G/U], where we define (g, t) hu := ghtu G/U. This element is well-defined since T normalizes U. Its dual K[G/U] = Hom K (K[G/U], K) can be identified with K[U \G]: indeed, let {e gu } be the canonical basis of K[G/U], and let {e gu } be the dual basis. For any element h G, and t T, we have h e gu = e h 1 gu, t e gu = e gt 1 U. Now, we define K[G/U] K[U \G], e gu e Ug 1,

5 CHARACTERS OF SL 2 (F q) 5 this establishes then an isomorphism of (KΓ, KΓ)-bimodules K[G/U] K[U \G]. On applying the general method in 2.1, we obtain the following two functors: and R K : KT modkg mod, V K[G/U] KT V, R K : KG modkt mod, W K[U \G] KG W, called Harish-Chandra induction and restriction respectively. As before, we denote by R K (or R) and R K (or R) the induced map between the Grothendieck groups. Definition 2.3 (Cuspidal character). An irreducible character χ of G is called cuspidal if there is no character α of T such that < χ, R(α) > G 0, i.e., R(χ) = 0. Remark 2.4. For any non zero virtual character α of T, its Harish- Chandra induction R(α) is not cuspidal. It is also possible to give an alternative definition of the Harish-Chandra induction (or restriction). For this, let V be a KT -module, we write V B the KB-module induced by the natural projection B T. Finally, we define Ind G B V B = KG KB V B. Moreover, for W a KG-module, if we look at the submodule W U, this gives a KT -module structure on W. Proposition 2.5. Let V ( resp. W ) be a KT -module ( resp. be a KGmodule). Then R K V Ind G B V B, 2.3. Mackey Formula. Proposition 2.6. Let α, β K 0 (KT ), then with s β(t) = β(s 1 ts) and R K W W U < R(α), R(β) > G =< α, β > T + < α, s β > T Remark 2.7. Note that s 1 ts = t 1 for any t T, hence s β = β is the dual character of β. Once β is a linear character, we have β = β 1. Now let α be any irreducible (linear) character of T µ q 1, since B G is of index q + 1, we find deg(r(α)) = q + 1. Moreover

6 6 CHARACTERS OF SL 2 (F q) (1) If α 2 1, < R(α), R(α) > G =< R(α), R(α 1 ) > G =< α, α > T = 1. As a result, R(α) = R(α 1 ) are irreducible of degree q + 1. (2) If α = α 0 is the unique linear character of order 2 of T F q µ q 1, then < R(α 0 ), R(α 0 ) > G =< α 0, α 0 > T + < α 0, α 0 > T = 2. As a result, R(α 0 ) has two non isomorphic irreducible factors: R(α 0 ) = R + (α 0 ) R (α 0 ). In particular, R(α 0 ) = R + (α 0 ) + R (α 0 ), with R + (α 0 ) R (α 0 ). (3) If α = 1 T, we have equally < R(1 T ), R(1 T ) >= 2. By Frobenius reciprocity, we have the following natural isomorphism Hom KG (1 G, Ind G B 1 B ) Hom KB (Res G B 1 G, 1 B ) = Hom KB (1 B, 1 B ) As a result, R K (1 T ) contains a copy of 1 G of multiplicity 1. We will denote by St G the other irreducible factor of R K (1 T ) (called the Steinberg character). Hence R(1 T ) = 1 G + St G Since deg(1 G ) = 1, and deg(r(1 T )) = q + 1. Hence deg(st G ) = q. (4) If α / {β, β 1 }, then < R(α), R(β) > G = 0. In this way, we obtain 4 + q 3 G = SL 2 (F q ). 2 = q+5 2 nonisomorphic irreducible characters of Proposition 2.8. deg(r + (α 0 )) = deg(r (α 0 )) = q Drinfeld curve 3.1. Definition of Drinfeld curve. In this, F is a fixed algebraic closure of F q (recall that q = p r with p an odd prime), and let Y A 2 F = Spec(F[X, Y ]) be the affine curve defined by the following equation: XY q Y X q = 1. It is easy to verify that this curve is smooth, irreducible. Moreover, G = SL 2 (F q ) acts on A 2 F via g (x, y) = (ax + by, cx + dy) with ( ) a b g = G. c d This action stabilizes Y. µ q+1 acts on A 2 F by homotheties, which induces an action on Y.

7 CHARACTERS OF SL 2 (F q) 7 the Frobenius endomorphism stabilizes equally Y. F : A 2 F A2 F, (x, y) (xq, y q ) These three actions satisfy the following relations: for any g G = SL 2 (F q ) and any ξ µ q+1, g ξ = ξ g g F = F g. F ξ = ξ 1 F Now consider the monoid G (µ q+1 <F> mon ) where the action of F on µ q+1 is given by ξ ξ 1. The previous relations give then an action of G (µ q+1 <F> mon ) on Y. Proposition 3.1. With the notations as before. 1. The group G acts freely on Y. 2. The group µ q+1 acts freely on A 2 F, and therefore also on Y Interesting quotients The following proposition is very useful for the determination of quotients. Recall that for a quasi-projective variety V/K, which is endowed with an action of finite group Γ. Then the fppf-quotient V/Γ is representable. When V = Spec(A) is affine, then V/Γ is representable by Spec(A G ) (SGA1). Proposition 3.2. Let V and W be two smooth and quasi-projective irreducible varieties over a field k, α : V W be a morphism of k-varieties, and Γ a finite group acting on V. Suppose that the following properties are satisfied: (1) α: X Y is surjective; (2) For any v, v V ( k), α(v) = α(v ) if and only v and v are in the same Γ-orbit; (3) There exists x 0 V ( k) such that the induced map of α between the tangent spaces v 0 is surjective. Then α: V/Γ W induced by α is an isomorphism of varieties. Proof. By definition, V/Γ is the cokernel of the following double morphism Γ V pr 2 Φ V α W p ᾱ? V/Γ

8 8 CHARACTERS OF SL 2 (F q) The second condition implies then the existence of ᾱ, which is surjective and purely inseparable by (2). The last condition implies then α is separable, as a result, so is ᾱ. Therefore, ᾱ is proper, quasifinite, hence is finite. Moreover, ᾱ is birational. The conclusion follows then from the fact that the scheme Y is normal Consider the map γ : Y A 1 F, (x, y) xyq2 yx q2 which is µ q+1 <F>-equivariant. Proposition 3.3. The map γ : Y which is an isomorphism. A 1 F induces a map γ : Y/G A1 F Proof. For the first assertion, since the varieties are separated over F, we only need to verify the corresponding statement on the F-points. Let then (x, y) Y (F) any element, and ( ) a b g = G, c d then g (x, y) = (ax + by, cx + dy). Further more (ax + by) (cx + dy) q2 (cx + dy) (ax + by) q2 = (ax + by) (cx q2 + dy q2 ) (cx + dy)(ax q2 + by q2 ) = xy q2 yx q2. That is, γ(g (x, y)) = γ(x, y). Whence the existence of the morphism γ. To verify that γ is an isomorphism, we apply then the criteria (proposition 3.2) The morphism v : Y A 1 F {0}, (x, y) y is µ q+1 <F> mon -equivariant, where the action of µ q+1 on A 1 F {0} is given by ξ z = ξz. This map is constant on the U-orbits, and induces by passing to quotient a map v : Y/U A 1 F {0} which is again µ q+1 <F> mon -equivariant. Proposition 3.4. The map v is an isomorphism.

9 CHARACTERS OF SL 2 (F q) The morphism π : Y P 1 F P1 F (F q), (x, y) [x : y] is well-defined and is G < F > mon -equivariant, which induces clearly an isomorphism π : Y/µ q+1 P 1 F P1 F (F q) 3.3. Fixed points under certain Frobenius endomorphism. In order to apply the Lefschetz fixed point theorem, we will need the following results. Proposition 3.5. Let ξ µ q+1, we have Y ξf =. Proof. Since π : Y P 1 F P1 F (F q) is µ q+1 <F> mon -equivariant. Hence we only need to show that ( P 1 F P 1 F (F q) ) F =, which is clear. Proposition 3.6. Let ξ µ q+1. Then { Y ξf 2 0 if ξ 1 = q 3 q if ξ = Compactification. If we identify A 2 F as the open subset {[x : y : z] P 2 (F) z 0} P 2 F = Proj(F[X, Y, Z]) then the Zariski closure Y of Y in P 2 F is given by the equation The complement XY q Y X q = Z q+1 Y Y = {[0 : 1 : 0]} {[1 : a : 0] a F q } P 1 F (F q). The action of G (µ q+1 <F> mon ) on Y extends naturally to Y : ( ) a b g [x : y : z] = [ax + by : cx + dy : z] with g = SL c d 2 (F q ); ξ [x : y : z] = [ξx : ξy : z]; F [x : y : z] = [x q : y q : z q ]. These actions stabilize Y. Moreover, one shows that Y /F is smooth. Finally, consider the morphism π 0 : Y P 1 F, [x : y : z] [x : y]

10 10 CHARACTERS OF SL 2 (F q) By the definition of π 0, this map π 0 is well-defined, G-equivariant and surjective. Moreover, it is constant on µ q+1 -orbits, and therefore induces, after passing to the quotient, a morphism of varieties π 0 : Y /µ q+1 P 1 F Proposition 3.7. The morphism π 0 : Y /µ q+1 P 1 F is a G <F > monequivariant isomorphism, which extends naturally the isomorphism π considered in Deligne-Lusztig Induction 4.1. Notations and Definitions. We will use the following notations: p = an odd prime, and q = p r. Y/F q the Drinfeld curve as defined in the previous ; l a prime different from p; K/Q l a large enough finite extension of Q l containing G -th roots of unity; We note H c(y, K) the compact support l-adic cohomology with coefficient in K. We will briefly denote it by H c(y ). These groups inherit hence a structure of G (µ q+1 <F> mon )-module. Definition 4.1. If θ is a character of µ q+1, we set R (θ) = ( 1) i [ H i ] c(y ) Kµq+1 V θ i 0 with V θ the Kµ q+1 -module admitting the character θ. In this way, we get a morphism of groups R : K 0 (Kµ q+1 ) K 0 (KG) which will be called Deligne-Lusztig induction. Remark 4.2. Let V be a representation of G µ q+1, and for any linear character θ of µ q+1, let V (θ) be the component of V such that µ q+1 acts via θ. We have then the following decomposition of G-representations V = V (θ) θ µ q+1 Let K θ0 be the irreducible representation associated to the linear character θ 0. Then we find { 0 if θ0 θ; V (θ) Kµq+1 K θ0 V (θ) if θ 0 = θ. G

11 CHARACTERS OF SL 2 (F q) 11 In the following, we will use the following results in l-adic cohomology. Let X/F be a quasi-projective variety with an action of a monoid Γ via endomorphisms. Proposition 4.3. Let Γ be a finite subgroup normalized by Γ, then we have an isomorphism of Γ/ -modules H i c(x/ ) H i c(x) Proposition 4.4. Let s and u be two invertible elements of Γ such that su = us, s is of order prime to p, and u is of order a power of p, then Tr X(su) = Tr X s(u). Proposition 4.5. Suppose Γ is a finite group, and let T be a torus acting on X which commutes with the action of Γ, then H c(x) H c(x T ) 4.2. First properties. As the curve Y/F is affine and irreducible of dimension 1, we have H i c(y ) = 0, for i / {1, 2}. Hence R (θ) = [ H 1 ] c(y ) Kµq+1 V θ G [ H 2 c(y ) Kµq+1 V θ ]G Moreover, by Poincaré duality, we know that Hence H 2 c(y, K) H 0 (Y, K) 1 G µq+1. [ H 2 c (Y ) ] G µ q+1 = 1 G µ q+1 and we get Proposition 4.6. If θ is a non-trivial linear character of µ q+1, then R (θ) = [ H 1 c(y ) Kµq+1 V θ ] G In particular, R (θ) is a character of G. Proposition 4.7. Let θ be a virtual character of Kµ q+1. Then R (θ) = R (θ ) = R (θ). Proof. Denote by φ: µ q+1 µ q+1 the morphism ξ ξ 1, then θ = φ θ. Let φ H i c(y ) the (KG, Kµ q+1 )-bimodule on which the action of µ q+1 is twisted by φ. To show the first equality, we only need to prove that (3) H i c(y ) φ H i c(y ) as (KG, Kµ q+1 )-modules. Since the Frobenius F induces an automorphism F : H i c(y ) H i c(y )

12 12 CHARACTERS OF SL 2 (F q) such that F g = g F, and F ξ = ξ 1 F = ξ 1 F, we get in this way the isomorphism (3) as (KG, Kµ q+1 )-bimodule. To show the second equality, for any g G, we have R (θ 1 )(g) = Tr Y (g, ξ 1 )θ (ξ) q + 1 ξ µ q+1 Similarly, R (θ) (g) = = = 1 q q q + 1 ξ µ q+1 Tr Y (g, ξ)θ(ξ). Tr Y (g 1, ξ 1 )θ (ξ 1 ) ξ µ q+1 Tr Y (g 1, ξ 1 )θ(ξ). ξ µ q+1 Hence, we are reduced to show that Tr Y (g, ξ) = Tr Y (g 1, ξ 1 ), which follows from the fact that (g, ξ) is of finite order and the fact that Tr Y (g, ξ) Z. This finishes then the proof The character R (1). By definition, we have Since R (1) = [ H 1 c(y ) µ q+1 ] G [ H 2 c(y ) µ q+1 ] = [ H 1 c(y/µ q+1 ) ] G 1 G. Y = Y P 1 (F q ) with µ q+1 acts trivially on Y Y, we obtain [H c(y/µ q+1 )] G = [H c(p 1 F )] G [H c(p 1 (F q ))] G. Lemma 4.8. We have [H c(p 1 (F q ))] G = K[G/B] G = 1 G + St G, and [H c(p 1 F )] = 2 1 G. Proof. Since H 1 c(p 1, K) = 0, we get immediately the equality concerning P 1 F. For the finite set P1 (F q ), recall that the action of G on P 1 (F q ) is given by ( ) a b g [x : y] = [ax + by : cx + dy], g = G = SL c d 2 (F q ). This action is transitive, and the for x 0 = [1 : 0] P 1 (F q ), we have the isotropic subgroup is B. Hence we find P 1 (F q ) G/B, and thus H c(p 1 (F q )) = H 0 c(p 1 (F q )) K[G/B]. It remains to compute K[G/B], which is Ind G B1 B = R K (1 T ).

13 CHARACTERS OF SL 2 (F q) 13 The second equality concerning H 0 c(p 1 (F q )) follows from Remark 2.7 (3). As a corollary, we get and therefore R (1) = St G 1 G, [ H i c (Y/µ q+1 ) ] 1 G if i = 2, = St G if i = 1, 0 otherwise 4.4. Dimensions. Let ξ be a non-trivial element of µ q+1. Then Tr Y (ξ) = Tr Y ξ (1) since Y ξ =, we find Tr Y (ξ) = 0. Hence as a character of µ q+1, [H c(y )] µq+1 is a multiple of the character of the regular representation. So for any θ µ q+1, we have deg(r (θ)) = deg(r (1)) = q Cuspidality. Proposition 4.9. Let α and θ be characters of T F q µ q 1 and µ q+1 respectively, then < R(α), R (θ) > G = 0. Proof. We may assume that θ is a linear character. Since R (1) = St G 1 G, and R(1) = St G + 1 G, we have < R(1), R (1) >= 0. Moreover, if α does not contain 1 G, we have equally < R(α), R (1) >= 0 (see Remark 2.7). Therefore, we may assume that θ 1. In this case, we know that R (θ) = [ H 1 c(y ) KT K θ ]G is a character of G (i.e., not only a virtual character), therefore, we only need to prove this result when α = reg T is the character of the regular representation. Since B/U T we get Hence reg T B = K[T ] = K[B/U] = Ind B U (1 U ), R(reg T ) = Ind G B(reg T B ) = Ind G B(Ind B U 1 U ) = Ind G U (1 U ). < R (θ), R(reg T ) >=< R (θ) U, 1 U >= dim K (H 1 c(y ) U Kµq+1 K θ ). Since H 1 c(y ) U = H 1 c(y/u) = H 1 c(a 1 F {0}) = 1 µ q+1 Therefore, for θ a linear character different from 1 µq+1, the θ-isotropic part of H 1 c(y ) U is trivial. This finishes the proof.

14 14 CHARACTERS OF SL 2 (F q) 4.6. Mackey formula. Proposition Let θ and η be two virtual characters of µ q+1. We have < R (θ), R (η) > G =< θ, η > µq+1 + < θ, η > µq+1 We set Z = Y Y with the diagonal action of G, and denote by µ (1) q+1 (resp. µ (2) q+1 ) the µ q+1 µ q+1 -set with underlying set µ q+1 and such that, if ζ µ q+1 and (ξ, ξ ) µ q+1 µ q+1, then (ξ, ξ ) ζ = ξξ ζ (resp. (ξ, ξ ) ζ = ξ 1 ξ ζ). This defines two K[µ q+1 µ q+1 ]-modules. Lemma With the notations as before. Then [ [ [H c (Z/G)] µq+1 µ q+1 K[µ (1) q+1 ]µ ] + K[µ (2) q+1 ]. q+1 µ q+1 ]µ q+1 µ q+1 Proof. Define and Z 0 = {(x, y, z, t) Z xt yz = 0} Z 0 = {(x, y, z, t) Z xt yz 0} Then Z 0 and Z 0 are (G µ q+1 µ q+1 )-stable subvarieties of Z. As a result, [H c(z/g)] µq+1 µ q+1 = [H c(z 0 /G)] µq+1 µ q+1 + [H c(z 0 /G)] µq+1 µ q+1 To compute the first term, consider the following isomorphism of F q -varieties µ q+1 Y Z 0, (ξ, x, y) (x, y, ξx, ξy) which induces then an action of G µ q+1 µ q+1 on µ q+1 Y given by Hence Z 0 /G µ q+1 A 1, and (g, ξ, ξ ) (ζ, x, y) = (ξ 1 ξ ζ, g (x, y)) [H c(z 0 /G)] µq+1 µ q+1 = [ K[µ (2) q+1 ] ] µ q+1 µ q+1 For the second term, consider the variety V = { (u, a, b) (A {0}) A 2 u q+1 ab = 1 } and the morphism ν : Z 0 V, (x, y, z, t) (xt yz, xt q yz q, x q t y q z) One can prove that ν induces an isomorphism of F q -varieties (see [2] 4.2): ν : Z 0 /G V. Under this isomorphism, the action of µ q+1 µ q+1 on V is given by (ξ, ξ ) (u, a, b) = (ξξ u, ξξ 1 a, ξ 1 ξ b).

15 CHARACTERS OF SL 2 (F q) 15 On the other hand, we can also define an action of G m,f on V as follows: λ (u, a, b) = (u, λa, λ 1 b) and this action commutes with that of µ q+1 µ q+1. As a consequence, we get H c(z 0 /G) µq+1 µ q+1 H c(v ) µq+1 µ q+1 H c(v G m,f ) µq+1 µ q+1. Now V G m,f = µ q+1 {0} {0}. Therefore [H c(z 0 /G)] µq+1 µ q+1 = This gives the formula of the Lemma. [ K [ ]] µ (1) q+1. µ q+1 µ q+1 Proof of Proposition By Proposition 4.7, we have < R (θ), R (η) > G =< 1 G, R (θ) R (η) > G =< 1 G, R (θ ) R (η) > G On the other hand, we have ( R (θ ) R (η) ) G = (( H c (Y ) Kµq+1 K θ ) K ( H c (Y ) Kµq+1 K η )) G where = H c ((Y Y )/G) K[µq+1 µ q+1 ] K θ η θ η : µ q+1 µ q+1 K, (ξ, ξ ) θ (ξ)η(ξ ) According to the key formula (Lemma 4.11), we have [ [ [H c ((Y Y )/G)] µq+1 µ q+1 K[µ (1) q+1 ]µ ] + K[µ (2) q+1 ]. q+1 µ q+1 ]µ q+1 µ q+1 Let µ (1) = {(ξ, ξ 1 ) ξ µ q+1 } and µ (2) = {(ξ, ξ) ξ µ q+1 }. Then [ ] K µ (i) q+1 = Ind µ q+1 µ q+1 1 µ (i) µ (i). Hence ( ) < R (θ), R (η) > G = dim K K[µ (1) q+1 ] K[µ q+1 µ q+1 ] K θ η ( ) +dim K K[µ (2) q+1 ] K[µ q+1 µ q+1 ] K θ η This finishes the proof. = < K θ η, Ind µ q+1 µ q+1 µ (1) 1 µ (1) > µq+1 µ q+1 + < K θ η, Ind µ q+1 µ q+1 µ (2) 1 µ (1) > µq+1 µ q+1 = < θ η, 1 µq > µq+1 + < θ η, 1 µq > µq = < θ, η > µq+1 + < θ, η > µq+1.

16 16 CHARACTERS OF SL 2 (F q) 4.7. Parametrization of Irr(G). By Mackey formula (Proposition 4.10) and 4.3, we have 1. R (1) = 1 G + St G ; 2. R (θ) = R (θ 1 ) Irr(G) if θ µ q+1 such that θ R (θ 0 ) = R +(θ 0 ) + R (θ 0 ), where R ±(θ 0 ) Irr(G) and R +(θ 0 ) R (θ 0 ). 4. If θ 2 1, η 2 1 and θ / {η, η 1 }, then R (θ) R (η). Remark In this way, we obtain (q 1)/2 + 2 = (q + 3)/2 cuspidal irreducible characters. Indeed, by Proposition 4.6, for any linear character θ 1, R (θ) is a character. Moreover, by definition, for any linear character α of T µ q 1, R(α) is a character. As a result, by the orthogonality property (Proposition 4.9), we have < R(α), R (θ) > G = 0. Hence, each irreducible component of R (θ) is different from those obtained in Remark 2.7. Hence by combining with Remark 2.7, we obtain q q = q + 4. On the other hand, as Irr(G) = q + 4, we conclude that these give all the characters of G = SL 2 (F q ). Proposition We have deg(r ±(θ 0 )) = q 1 2. Proof. Let d + = deg(r +(θ 0 )), and d = deg(r (θ 0 )). Then G = q 2 + q 3 ( ) q (q + 1) q (q 1)2 + d d 2. Moreover, d + + d = deg(r (θ 0 )) = q 1. In this way, we get d + = d = q Action of the Frobenius Endomorphism. Recall that the Frobenius F acts by automorphism on H i c(y ), and is given by multiplication by q when i = 2. Let now θ be a linear character of µ q+1, and e θ = 1 θ(ξ 1 )ξ Kµ q+1. q + 1 ξ µ q+1

17 CHARACTERS OF SL 2 (F q) 17 Then for the Kµ q+1 -module H i c(y ), we have the following isomorphism H i c(y ) Kµq+1 K θ H i c(y )e θ H i c(y ) Since F ξ = ξ 1 F, and F is an automorphism of H i, as a result, we find F (H i c(y )e θ ) = H i c(y ) eθ 1. In particular, F stabilizes H 1 c(y )e 1 and H 1 c(y )e θ Action of F on H 1 c(y )e 1. The KG-module H 1 c(y )e 1 = H 1 c(y ) µ q+1 = H 1 c(y/µ q+1 ) St G is irreducible, and since F commutes with the action of G, F acts H 1 c(y )e 1 as multiplication by a scalar ρ 1. By the Lefschetez fixed point theorem (ref...), we have 0 = (P 1 (F) P 1 (F q )) F = q ρ1 dim K (H 1 c(y/µ q+1 )) = q(1 ρ 1 ) As a result, ρ 1 = Action of F on H 1 c(y )e θ0. Set H 1 c(y )e θ0 = V + θ 0 V θ 0 with V ± θ 0 the irreducible factors of H 1 c(y )e θ0 with characters R (θ 0 ) ±. As a result, F acts on V ± θ 0 by multiplication of scalars ρ ±. Moreover, since V ± θ 0 are two non isomorphic representations, we find in this way F = ρ + ρ. By Lefschetz fixed-point theorem, we have 0 = Y (F) F = q qρ1 (q 1)(ρ + + ρ 1 ) Tr ( F, 2 θ:θ2 1H 1 ) c(y )e θ As a result, we find ρ + = ρ. To explicitly calculate ρ ±, we will study the action of F 2. As F 2 stabilizes H 1 c(y )e θ, and it acts by multiplication by a scalar λ θ. Hence λ θ0 = ρ 2 ±. Theorem Let θ be a linear character of µ q+1. then { 1 if θ = 1, λ θ = θ( 1)q if θ 1. Proof. The equality λ 1 = 1 has been verified. By the Lefechetz fixed-point theorem, we have Y ξf 2 = q 2 qλ 1 1)θ(ξ)λ θ θ 1(q for all ξ µ q+1. Hence Y ξf 2 = (q 2 1) (q 1) (q 1)θ(ξ)λ θ θ µ q+1

18 18 CHARACTERS OF SL 2 (F q) As a result, θ µ q+1 θ(ξ)λ θ = (q + 1) Y ξf { 2 1 q q 1 = 2 if ξ = 1, q + 1 if ξ 1. These give then a system equations, which admits a unique solution. One can verify that the numbers λ θ give the unique solution. This finishes the proof. Corollary We have ρ ± = ± θ 0 ( 1)q Action on H 1 c(y ) eθ H 1 c(y ). eθ 1 Suppose θ µ q+1 θ 2 1. Since F stabilizes such that and sends H 1 c(y ) eθ each of which has multiplicity q 1. H 1 c(y ) eθ H 1 c(y ) eθ 1 onto H 1 c(y ) eθ 1, therefore F has two eigenvalues θ( 1)q, θ( 1)q References [1] M. Isaacs. Character theory of finite groups. Dover Publications, INC. New York [2] C. Bonnafé. Reprentations of SL 2 (F q ). Algebra and Applications, 13. Springer.

THREE CASES AN EXAMPLE: THE ALTERNATING GROUP A 5

THREE CASES AN EXAMPLE: THE ALTERNATING GROUP A 5 THREE CASES REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE LECTURE II: DELIGNE-LUSZTIG THEORY AND SOME APPLICATIONS Gerhard Hiss Lehrstuhl D für Mathematik RWTH Aachen University Summer School Finite Simple

More information

Character Sheaves and GGGRs

Character Sheaves and GGGRs Character Sheaves and GGGRs Jay Taylor Technische Universität Kaiserslautern Algebra Seminar University of Georgia 24th March 2014 Jay Taylor (TU Kaiserslautern) Character Sheaves Georgia, March 2014 1

More information

`-modular Representations of Finite Reductive Groups

`-modular Representations of Finite Reductive Groups `-modular Representations of Finite Reductive Groups Bhama Srinivasan University of Illinois at Chicago AIM, June 2007 Bhama Srinivasan (University of Illinois at Chicago) Modular Representations AIM,

More information

Odds and ends on equivariant cohomology and traces

Odds and ends on equivariant cohomology and traces Odds and ends on equivariant cohomology and traces Weizhe Zheng Columbia University International Congress of Chinese Mathematicians Tsinghua University, Beijing December 18, 2010 Joint work with Luc Illusie.

More information

Character Sheaves and GGGRs

Character Sheaves and GGGRs and GGGRs Jay Taylor Technische Universität Kaiserslautern Global/Local Conjectures in Representation Theory of Finite Groups Banff, March 2014 Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March

More information

Notes on Green functions

Notes on Green functions Notes on Green functions Jean Michel University Paris VII AIM, 4th June, 2007 Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 1 / 15 We consider a reductive group G over

More information

0 A. ... A j GL nj (F q ), 1 j r

0 A. ... A j GL nj (F q ), 1 j r CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected

More information

Geometric Structure and the Local Langlands Conjecture

Geometric Structure and the Local Langlands Conjecture Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure

More information

A PROOF OF BURNSIDE S p a q b THEOREM

A PROOF OF BURNSIDE S p a q b THEOREM A PROOF OF BURNSIDE S p a q b THEOREM OBOB Abstract. We prove that if p and q are prime, then any group of order p a q b is solvable. Throughout this note, denote by A the set of algebraic numbers. We

More information

LECTURE 11: SOERGEL BIMODULES

LECTURE 11: SOERGEL BIMODULES LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing

More information

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated

More information

Category O and its basic properties

Category O and its basic properties Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α

More information

CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0. Mitya Boyarchenko Vladimir Drinfeld. University of Chicago

CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0. Mitya Boyarchenko Vladimir Drinfeld. University of Chicago CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0 Mitya Boyarchenko Vladimir Drinfeld University of Chicago Some historical comments A geometric approach to representation theory for unipotent

More information

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F)

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) IVAN LOSEV In this lecture we will discuss the representation theory of the algebraic group SL 2 (F) and of the Lie algebra sl 2 (F), where F is

More information

l-adic Representations

l-adic Representations l-adic Representations S. M.-C. 26 October 2016 Our goal today is to understand l-adic Galois representations a bit better, mostly by relating them to representations appearing in geometry. First we ll

More information

Equivariant Algebraic K-Theory

Equivariant Algebraic K-Theory Equivariant Algebraic K-Theory Ryan Mickler E-mail: mickler.r@husky.neu.edu Abstract: Notes from lectures given during the MIT/NEU Graduate Seminar on Nakajima Quiver Varieties, Spring 2015 Contents 1

More information

CHARACTER SHEAVES, TENSOR CATEGORIES AND NON-ABELIAN FOURIER TRANSFORM

CHARACTER SHEAVES, TENSOR CATEGORIES AND NON-ABELIAN FOURIER TRANSFORM CHARACTER SHEAVES, TENSOR CATEGORIES AND NON-ABELIAN FOURIER TRANSFORM Abstract. Notes from a course given by Victor Ostrik in Luminy, 2. Notes by Geordie Williamson.. Character sheaves Character sheaves

More information

A partition of the set of enhanced Langlands parameters of a reductive p-adic group

A partition of the set of enhanced Langlands parameters of a reductive p-adic group A partition of the set of enhanced Langlands parameters of a reductive p-adic group joint work with Ahmed Moussaoui and Maarten Solleveld Anne-Marie Aubert Institut de Mathématiques de Jussieu - Paris

More information

TCC Homological Algebra: Assignment #3 (Solutions)

TCC Homological Algebra: Assignment #3 (Solutions) TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate

More information

Representations Are Everywhere

Representations Are Everywhere Representations Are Everywhere Nanghua Xi Member of Chinese Academy of Sciences 1 What is Representation theory Representation is reappearance of some properties or structures of one object on another.

More information

Notes on p-divisible Groups

Notes on p-divisible Groups Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

More information

Math 249B. Geometric Bruhat decomposition

Math 249B. Geometric Bruhat decomposition Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique

More information

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

More information

EQUIVARIANCE AND EXTENDIBILITY IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER

EQUIVARIANCE AND EXTENDIBILITY IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER EQUIVARIANCE AND EXTENDIBILITY IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER MARC CABANES AND BRITTA SPÄTH Abstract. We show that several character correspondences for finite reductive groups G are

More information

Thus we get. ρj. Nρj i = δ D(i),j.

Thus we get. ρj. Nρj i = δ D(i),j. 1.51. The distinguished invertible object. Let C be a finite tensor category with classes of simple objects labeled by a set I. Since duals to projective objects are projective, we can define a map D :

More information

HEIGHT 0 CHARACTERS OF FINITE GROUPS OF LIE TYPE

HEIGHT 0 CHARACTERS OF FINITE GROUPS OF LIE TYPE REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 00, Pages 000 000 (Xxxx XX, XXXX) S 1088-4165(XX)0000-0 HEIGHT 0 CHARACTERS OF FINITE GROUPS OF LIE TYPE GUNTER MALLE

More information

REPRESENTATION THEORY. WEEK 4

REPRESENTATION THEORY. WEEK 4 REPRESENTATION THEORY. WEEK 4 VERA SERANOVA 1. uced modules Let B A be rings and M be a B-module. Then one can construct induced module A B M = A B M as the quotient of a free abelian group with generators

More information

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n)

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) VERA SERGANOVA Abstract. We decompose the category of finite-dimensional gl (m n)- modules into the direct sum of blocks, show that

More information

Topic Proposal Applying Representation Stability to Arithmetic Statistics

Topic Proposal Applying Representation Stability to Arithmetic Statistics Topic Proposal Applying Representation Stability to Arithmetic Statistics Nir Gadish Discussed with Benson Farb 1 Introduction The classical Grothendieck-Lefschetz fixed point formula relates the number

More information

Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)]. Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

More information

SIMPLE GROUPS ADMIT BEAUVILLE STRUCTURES

SIMPLE GROUPS ADMIT BEAUVILLE STRUCTURES SIMPLE GROUPS ADMIT BEAUVILLE STRUCTURES ROBERT GURALNICK AND GUNTER MALLE Dedicated to the memory of Fritz Grunewald Abstract. We prove a conjecture of Bauer, Catanese and Grunewald showing that all finite

More information

The Major Problems in Group Representation Theory

The Major Problems in Group Representation Theory The Major Problems in Group Representation Theory David A. Craven 18th November 2009 In group representation theory, there are many unsolved conjectures, most of which try to understand the involved relationship

More information

Representation Theory

Representation Theory Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section II 19I 93 (a) Define the derived subgroup, G, of a finite group G. Show that if χ is a linear character

More information

SOLVABLE FUSION CATEGORIES AND A CATEGORICAL BURNSIDE S THEOREM

SOLVABLE FUSION CATEGORIES AND A CATEGORICAL BURNSIDE S THEOREM SOLVABLE FUSION CATEGORIES AND A CATEGORICAL BURNSIDE S THEOREM PAVEL ETINGOF The goal of this talk is to explain the classical representation-theoretic proof of Burnside s theorem in finite group theory,

More information

REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES. Notation. 1. GL n

REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES. Notation. 1. GL n REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES ZHIWEI YUN Fix a prime number p and a power q of p. k = F q ; k d = F q d. ν n means ν is a partition of n. Notation Conjugacy classes 1. GL n 1.1.

More information

SEMISIMPLE SYMPLECTIC CHARACTERS OF FINITE UNITARY GROUPS

SEMISIMPLE SYMPLECTIC CHARACTERS OF FINITE UNITARY GROUPS SEMISIMPLE SYMPLECTIC CHARACTERS OF FINITE UNITARY GROUPS BHAMA SRINIVASAN AND C. RYAN VINROOT Abstract. Let G = U(2m, F q 2) be the finite unitary group, with q the power of an odd prime p. We prove that

More information

Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

More information

A PROOF OF BOREL-WEIL-BOTT THEOREM

A PROOF OF BOREL-WEIL-BOTT THEOREM A PROOF OF BOREL-WEIL-BOTT THEOREM MAN SHUN JOHN MA 1. Introduction In this short note, we prove the Borel-Weil-Bott theorem. Let g be a complex semisimple Lie algebra. One basic question in representation

More information

Math 594. Solutions 5

Math 594. Solutions 5 Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses

More information

On the modular curve X 0 (23)

On the modular curve X 0 (23) On the modular curve X 0 (23) René Schoof Abstract. The Jacobian J 0(23) of the modular curve X 0(23) is a semi-stable abelian variety over Q with good reduction outside 23. It is simple. We prove that

More information

PROBLEMS, MATH 214A. Affine and quasi-affine varieties

PROBLEMS, MATH 214A. Affine and quasi-affine varieties PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset

More information

DUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE

DUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE DUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE C. RYAN VINROOT Abstract. We prove that the duality operator preserves the Frobenius- Schur indicators of characters

More information

FUNCTORS AND ADJUNCTIONS. 1. Functors

FUNCTORS AND ADJUNCTIONS. 1. Functors FUNCTORS AND ADJUNCTIONS Abstract. Graphs, quivers, natural transformations, adjunctions, Galois connections, Galois theory. 1.1. Graph maps. 1. Functors 1.1.1. Quivers. Quivers generalize directed graphs,

More information

REPRESENTATION THEORY WEEK 5. B : V V k

REPRESENTATION THEORY WEEK 5. B : V V k REPRESENTATION THEORY WEEK 5 1. Invariant forms Recall that a bilinear form on a vector space V is a map satisfying B : V V k B (cv, dw) = cdb (v, w), B (v 1 + v, w) = B (v 1, w)+b (v, w), B (v, w 1 +

More information

SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

More information

LECTURE 4: SOERGEL S THEOREM AND SOERGEL BIMODULES

LECTURE 4: SOERGEL S THEOREM AND SOERGEL BIMODULES LECTURE 4: SOERGEL S THEOREM AND SOERGEL BIMODULES DMYTRO MATVIEIEVSKYI Abstract. These are notes for a talk given at the MIT-Northeastern Graduate Student Seminar on category O and Soergel bimodules,

More information

8 Perverse Sheaves. 8.1 Theory of perverse sheaves

8 Perverse Sheaves. 8.1 Theory of perverse sheaves 8 Perverse Sheaves In this chapter we will give a self-contained account of the theory of perverse sheaves and intersection cohomology groups assuming the basic notions concerning constructible sheaves

More information

Riemann surfaces with extra automorphisms and endomorphism rings of their Jacobians

Riemann surfaces with extra automorphisms and endomorphism rings of their Jacobians Riemann surfaces with extra automorphisms and endomorphism rings of their Jacobians T. Shaska Oakland University Rochester, MI, 48309 April 14, 2018 Problem Let X be an algebraic curve defined over a field

More information

Adic Spaces. Torsten Wedhorn. June 19, 2012

Adic Spaces. Torsten Wedhorn. June 19, 2012 Adic Spaces Torsten Wedhorn June 19, 2012 This script is highly preliminary and unfinished. It is online only to give the audience of our lecture easy access to it. Therefore usage is at your own risk.

More information

A short proof of Klyachko s theorem about rational algebraic tori

A short proof of Klyachko s theorem about rational algebraic tori A short proof of Klyachko s theorem about rational algebraic tori Mathieu Florence Abstract In this paper, we give another proof of a theorem by Klyachko ([?]), which asserts that Zariski s conjecture

More information

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture I

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture I Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture I Set-up. Let K be an algebraically closed field. By convention all our algebraic groups will be linear algebraic

More information

LECTURE 3: TENSORING WITH FINITE DIMENSIONAL MODULES IN CATEGORY O

LECTURE 3: TENSORING WITH FINITE DIMENSIONAL MODULES IN CATEGORY O LECTURE 3: TENSORING WITH FINITE DIMENSIONAL MODULES IN CATEGORY O CHRISTOPHER RYBA Abstract. These are notes for a seminar talk given at the MIT-Northeastern Category O and Soergel Bimodule seminar (Autumn

More information

Math 210C. The representation ring

Math 210C. The representation ring Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

More information

Algebraic group actions and quotients

Algebraic group actions and quotients 23rd Autumn School in Algebraic Geometry Algebraic group actions and quotients Wykno (Poland), September 3-10, 2000 LUNA S SLICE THEOREM AND APPLICATIONS JEAN MARC DRÉZET Contents 1. Introduction 1 2.

More information

15 Elliptic curves and Fermat s last theorem

15 Elliptic curves and Fermat s last theorem 15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine

More information

1 Notations and Statement of the Main Results

1 Notations and Statement of the Main Results An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

More information

New York Journal of Mathematics. Cohomology of Modules in the Principal Block of a Finite Group

New York Journal of Mathematics. Cohomology of Modules in the Principal Block of a Finite Group New York Journal of Mathematics New York J. Math. 1 (1995) 196 205. Cohomology of Modules in the Principal Block of a Finite Group D. J. Benson Abstract. In this paper, we prove the conjectures made in

More information

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when

More information

Representations and Linear Actions

Representations and Linear Actions Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

More information

Wild ramification and the characteristic cycle of an l-adic sheaf

Wild ramification and the characteristic cycle of an l-adic sheaf Wild ramification and the characteristic cycle of an l-adic sheaf Takeshi Saito March 14 (Chicago), 23 (Toronto), 2007 Abstract The graded quotients of the logarithmic higher ramification groups of a local

More information

Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves. Yuichiro Hoshi

Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves. Yuichiro Hoshi Hokkaido Mathematical Journal ol. 45 (2016) p. 271 291 Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves uichiro Hoshi (Received February 28, 2014; Revised June 12, 2014) Abstract.

More information

REPRESENTATION THEORY, LECTURE 0. BASICS

REPRESENTATION THEORY, LECTURE 0. BASICS REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite

More information

Symplectic varieties and Poisson deformations

Symplectic varieties and Poisson deformations Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the

More information

We then have an analogous theorem. Theorem 1.2.

We then have an analogous theorem. Theorem 1.2. 1. K-Theory of Topological Stacks, Ryan Grady, Notre Dame Throughout, G is sufficiently nice: simple, maybe π 1 is free, or perhaps it s even simply connected. Anyway, there are some assumptions lurking.

More information

Some remarks on Frobenius and Lefschetz in étale cohomology

Some remarks on Frobenius and Lefschetz in étale cohomology Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)

More information

Lemma 1.3. The element [X, X] is nonzero.

Lemma 1.3. The element [X, X] is nonzero. Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

More information

Introduction to Chiral Algebras

Introduction to Chiral Algebras Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument

More information

Smooth morphisms. Peter Bruin 21 February 2007

Smooth morphisms. Peter Bruin 21 February 2007 Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,

More information

SIX LECTURES ON DELIGNE-LUSZTIG THEORY

SIX LECTURES ON DELIGNE-LUSZTIG THEORY SIX LECTURES ON DELIGNE-LUSZTIG THEORY Abstract. Lectures given by Raphael Rouquier in Hilary term, Oxford, 2010. Notes of lectures 2, 3, 4 and 6 by Geordie Williamson, lecture 5 by David Craven. 1. Lecture

More information

SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS

SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the

More information

CURTIS HOMOMORPHISMS AND THE INTEGRAL BERNSTEIN CENTER FOR GL n

CURTIS HOMOMORPHISMS AND THE INTEGRAL BERNSTEIN CENTER FOR GL n CURTIS HOMOMORPHISMS AND THE INTEGRAL BERNSTEIN CENTER OR GL n DAVID HELM Abstract. We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center

More information

ORAL QUALIFYING EXAM QUESTIONS. 1. Algebra

ORAL QUALIFYING EXAM QUESTIONS. 1. Algebra ORAL QUALIFYING EXAM QUESTIONS JOHN VOIGHT Below are some questions that I have asked on oral qualifying exams (starting in fall 2015). 1.1. Core questions. 1. Algebra (1) Let R be a noetherian (commutative)

More information

DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY C.M. RINGEL)

DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY C.M. RINGEL) DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY CM RINGEL) C BOWMAN, SR DOTY, AND S MARTIN Abstract We give an algorithm for working out the indecomposable

More information

The Diamond Category of a Locally Discrete Ordered Set.

The Diamond Category of a Locally Discrete Ordered Set. The Diamond Category of a Locally Discrete Ordered Set Claus Michael Ringel Let k be a field Let I be a ordered set (what we call an ordered set is sometimes also said to be a totally ordered set or a

More information

Iwasawa algebras and duality

Iwasawa algebras and duality Iwasawa algebras and duality Romyar Sharifi University of Arizona March 6, 2013 Idea of the main result Goal of Talk (joint with Meng Fai Lim) Provide an analogue of Poitou-Tate duality which 1 takes place

More information

H(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).

H(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )). 92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported

More information

Brauer Theory for Profinite Groups

Brauer Theory for Profinite Groups Brauer Theory for Profinite Groups John MacQuarrie 1, Peter Symonds Abstract Brauer Theory for a finite group can be viewed as a method for comparing the representations of the group in characteristic

More information

Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms

Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms U-M Automorphic forms workshop, March 2015 1 Definition 2 3 Let Γ = PSL 2 (Z) Write ( 0 1 S =

More information

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

More information

A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander

A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander During the first three days of September, 1997, I had the privilege of giving a series of five lectures at the beginning of the School on Algebraic

More information

Modular representation theory

Modular representation theory Modular representation theory 1 Denitions for the study group Denition 1.1. Let A be a ring and let F A be the category of all left A-modules. The Grothendieck group of F A is the abelian group dened by

More information

On The Mackey Formula for Connected Centre Groups Jay Taylor

On The Mackey Formula for Connected Centre Groups Jay Taylor On The Mackey Formula for Connected Centre Groups Jay Taylor Abstract. Let G be a connected reductive algebraic group over F p and let F : G G be a Frobenius endomorphism endowing G with an F q -rational

More information

Real and p-adic Picard-Vessiot fields

Real and p-adic Picard-Vessiot fields Spring Central Sectional Meeting Texas Tech University, Lubbock, Texas Special Session on Differential Algebra and Galois Theory April 11th 2014 Real and p-adic Picard-Vessiot fields Teresa Crespo, Universitat

More information

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)

More information

CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0. Mitya Boyarchenko Vladimir Drinfeld. University of Chicago

CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0. Mitya Boyarchenko Vladimir Drinfeld. University of Chicago arxiv:1301.0025v1 [math.rt] 31 Dec 2012 CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0 Mitya Boyarchenko Vladimir Drinfeld University of Chicago Overview These are slides for a talk given

More information

1. Algebraic vector bundles. Affine Varieties

1. Algebraic vector bundles. Affine Varieties 0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

More information

Marko Tadić. Introduction Let F be a p-adic field. The normalized absolute value on F will be denoted by F. Denote by ν : GL(n, F ) R the character

Marko Tadić. Introduction Let F be a p-adic field. The normalized absolute value on F will be denoted by F. Denote by ν : GL(n, F ) R the character ON REDUCIBILITY AND UNITARIZABILITY FOR CLASSICAL p-adic GROUPS, SOME GENERAL RESULTS Marko Tadić Abstract. The aim of this paper is to prove two general results on parabolic induction of classical p-adic

More information

NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES

NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES KENTA UEYAMA Abstract. Gorenstein isolated singularities play an essential role in representation theory of Cohen-Macaulay modules. In this article,

More information

Lie Algebra Cohomology

Lie Algebra Cohomology Lie Algebra Cohomology Carsten Liese 1 Chain Complexes Definition 1.1. A chain complex (C, d) of R-modules is a family {C n } n Z of R-modules, together with R-modul maps d n : C n C n 1 such that d d

More information

On the geometric Langlands duality

On the geometric Langlands duality On the geometric Langlands duality Peter Fiebig Emmy Noether Zentrum Universität Erlangen Nürnberg Schwerpunkttagung Bad Honnef April 2010 Outline This lecture will give an overview on the following topics:

More information

Weil Conjectures (Deligne s Purity Theorem)

Weil Conjectures (Deligne s Purity Theorem) Weil Conjectures (Deligne s Purity Theorem) David Sherman, Ka Yu Tam June 7, 2017 Let κ = F q be a finite field of characteristic p > 0, and k be a fixed algebraic closure of κ. We fix a prime l p, and

More information

descends to an F -torus S T, and S M since S F ) 0 red T F

descends to an F -torus S T, and S M since S F ) 0 red T F Math 249B. Basics of reductivity and semisimplicity In the previous course, we have proved the important fact that over any field k, a non-solvable connected reductive group containing a 1-dimensional

More information

Reducibility of generic unipotent standard modules

Reducibility of generic unipotent standard modules Journal of Lie Theory Volume?? (??)???? c?? Heldermann Verlag 1 Version of March 10, 011 Reducibility of generic unipotent standard modules Dan Barbasch and Dan Ciubotaru Abstract. Using Lusztig s geometric

More information

Algebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.

More information

ON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig

ON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig ON SOME PARTITIONS OF A FLAG MANIFOLD G. Lusztig Introduction Let G be a connected reductive group over an algebraically closed field k of characteristic p 0. Let W be the Weyl group of G. Let W be the

More information

Bruhat Tits buildings and representations of reductive p-adic groups

Bruhat Tits buildings and representations of reductive p-adic groups Bruhat Tits buildings and representations of reductive p-adic groups Maarten Solleveld Radboud Universiteit Nijmegen joint work with Ralf Meyer 26 November 2013 Starting point Let G be a reductive p-adic

More information

Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society

Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society Grothendieck Messing deformation theory for varieties of K3 type Andreas Langer and Thomas Zink

More information

Primitive Ideals and Unitarity

Primitive Ideals and Unitarity Primitive Ideals and Unitarity Dan Barbasch June 2006 1 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H)

More information

Notes 10: Consequences of Eli Cartan s theorem.

Notes 10: Consequences of Eli Cartan s theorem. Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation

More information

ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

More information