Notes on representations of finite groups

Transcription

1 Notes on representations of finite groups Mark Reeder November 30, 2014 Contents 1 Basic notions 4 1.1, and Hom Invariant vectors Invariant subspaces and irreducible representations The group algebra Dependence on the ground field Permutation representations Interwining algebras of permutation representations Examples: the reflection representations of symmetric and general linear groups 9 2 Semisimplicity and characters Projecting onto the invariants Projecting onto the equivariants Maschke s Theorem Schur s Lemma Interwining and irreducibility Multiplicity Isotypic components and multiplicity one

2 2.7 The Regular Representation Permutation representations again The number of irreducible representations Induced Representations Conjugate representations Definition and first properties of induced representations Induction and Restriction Representations of GL 2 (q) The Gelfand-Graev representation Principal series representations Cuspidal representations Representations of S n Exterior power representations Grassmannian representations General irreducible representations of S n The representations of S More on intertwining algebras Multiplicity-free representations Representations of GL 2 (q), again Other interpretations of induced representations Tensor products Vector bundles Example: Principal series of SL 2 (q) Example: Polynomial representations of SL 2 (C)

3 7.5 Clifford s theorem Example: Non-abelian groups of order p Characters of SL 2 (q) The group SL 2 (q) and some of its subgroups Conjugacy classes in SL 2 (q) When q is odd λ = λ 1 {±1} λ λ 1 f λ λ 1 / f Checking the results When q is even λ = λ 1 = λ λ 1 f λ λ 1 / f Checking the results Irreducible principal series representations Reducible principal series representations Cuspidal representations Reducible π η The character table for q even The character table for q odd Example: q = Weil representations of SL 2 (3) Example: q = P SL 2 (q) P GL 2 (q)

4 9 Integrality and the center of the group algebra The ring of algebraic integers The center of the group algebra The dimension of an irreducible representation Zeros of characters and Burnsides p a q b theorem Basic notions Let G be a group and let F be a field. An F -representation of G is a pair (ρ, V ), where ρ : G GL(V ), is a homomorphism from G to the group GL(V ) of automorphisms of an F -vector space V. We often abbreviate the pair (ρ, V ) by just ρ or V, when it is not necessary to specify the other ingredient. If (ρ, V ) and (ρ, V ) are two F -representations of G, a vector space homomorphim T : V V such that ρ (g) T = T ρ(g) for all g G is called G-equivariant or an intertwining map. We let Hom G (V, V ) denote the F -vector space of G equivariant maps T Hom F (V, V ). An isomorphism of F -representations (ρ, V ) and (ρ, V ) is a G-equivariant linear isomorphism T : V V. Thus, we have a category Rep F (G), whose objects are the F -representations of G, with morphisms Hom G (V, V ). If V is finite dimensional and we choose a basis of V, then an F -representation of G on V may be viewed as a homomorphism ρ : G GL n (F ). If we choose another basis of V we get another homomorphism ρ : G GL n (F ) such that ρ(g) = T ρ (g)t 1, where T GL n (F ) is the transition matrix between the two bases. 1.1, and Hom Certain functors on vector spaces extend to representations. Given two F -representations (ρ, V ) and (ρ, V ) of G we can form new F representations of G via ρ ρ (g) = ρ(g) ρ (g) GL(V V ), ρ ρ (g) = ρ(g) ρ (g) GL(V F V ). 4

5 We also have a representation ρ : G GL(Hom F (V, V )) defined by [ρ (g)f](g) = ρ (g) f ρ(g) Invariant vectors Let W be a one-dimensional vector space over F. The trivial representation is the one dimensional representation 1 G : G GL 1 (F ) = F given by 1 G (g) = 1 for all g. If (ρ, V ) is a general F - representation, the subspace is the space of G-invariants in V. Note that as F -vector spaces. V G := {v V : ρ(g)v = v for all g G} V G Hom G (1 G, V ) Invariants are important even for non-trivial representations. For example, given two representations (ρ, V ) and (ρ, V ), the invariants in Hom F (V, V ) are exactly the intertwining maps: Hom F (V, V ) G = Hom G (V, V ). 1.3 Invariant subspaces and irreducible representations Let (ρ, V ) be an F -representation of G. A G-invariant subspace of V is an F - subspace U V such that ρ(g)u = U. Equivalently U is G-invariant precisely when the inclusion map U V belongs to Hom G (U, V ). Note that if U is an invariant subspace then restricting each ρ(g) to U gives a representation ρ : G GL(U). We say V is irreducible if the only G-invariant subspaces are 0 and V itself. We let Irr F (G) be the set of isomorphism classes of irreducible F -representations of G. Much of Representation Theory is about understanding Irr F (G) for a given group G and field F. 1.4 The group algebra An F -representation ρ : G GL(V ) extends to a ring homomorphism ( ) ρ : F [G] End(V ), ρ c g g = c g ρ(g) End(V ), g G g G 5

6 whose image is the subalgebra of End(V ) generated by ρ(g). The correspondence ρ ρ is an equivalence Rep F (G) Mod(F [G]) between the categories of F -representations of G and F [G]-modules. Under this equivalence, G- invariant subspaces correspond to F [G]-submodules and irreducible representations of G correspond to simple F [G]-modules. 1.5 Dependence on the ground field We give an example to show that Irr F (G) depends significantly on the field F. Let G = C p be a cyclic group of prime order p and choose a generator g C p. The group algebra F [G] is a quotient of the polynomial ring F [x], namely F [G] F [x]/(x p 1), g x. Hence the structure of F [G] depends on how x p 1 factors in F [x]. If F = C then letting ζ = e 2πi/p, we have p 1 C[G] C[x]/(x ζ i ), i=0 so Irr C (G) = {ρ 0,..., ρ p 1 }, where each ρ i is a one-dimensional C-representation of G on which ρ i (g) is multiplication by ζ i. If F = Q then Q[G] Q[x]/(x 1) Q[x]/(Φ p ), where Φ p = 1 + x + + x p 1 is irreducible in Q[x]. Hence Irr Q (G) = {1 G, ρ}, where ρ(g) is multiplication by x on the p 1-dimensional Q-vector space V = Q[x]/(Φ p ). If we use the basis {1, x,..., x p 2 } of V then ρ(g) = C Φp is the companion matrix of Φ p. If F = F p then F p [G] F p [x]/(x 1) p has but one simple module, namely Irr Fp (G) = {1 G }. This does not mean that the category Rep Fp (G) is uninteresting. Indeed, the F p -representations of dimension n are in one-to-one correspondence with conjugacy classes of elements of order p in GL n (p). 1.6 Permutation representations Suppose G acts on a set X. Let V X be the F -vector space of functions f : V F with finite support. Then we have a representation ρ : G GL(V X ) given by [ρ(g)f](x) = f(g 1 x). 6

7 For each x X let e x V X be the function e x (x ) = { 1 if x = x 0 if x x. In terms of the basis {e x : x X} we have ρ(g)e x = e g x. Suppose now that X is finite. Then the constant function f 0 1 belongs to VX G. More generally, for each G-orbit O X the function f O V X defined by { 1 if x O f O (x) = 0 if x / O is invariant under G and the invariant subspace V G X has basis {f O : O G\X}. In particular we have dim V G X = G\X. The subspace is G-invariant, so we have a representation V 0 X := {f V : f(x) = 0} x X ρ 0 : G GL(V 0 X). If X 0 in F then f 0 / VX 0 and we have a decomposition into a direct sum of G-invariant subspaces: V = V 0 X F f 0 V 0 X 1 G. If X = 0 in F then f 0 V 0 X and we have a chain of invariant subspaces F f 0 V 0 X V X with V 0 X of codimension one in V X Interwining algebras of permutation representations Suppose G acts transitively on the finite set X and that F has characteristic zero. Using the basis {e x } it is clear that tr(ρ(g), V X ) = X g is the size of the fixed-point set of g in X. This formula has a beautiful generalization (see (1) below, that I learned from work of G. Lusztig, though it probably goes back to Schur), via the interwining algebra H := End G (V X ). 7

8 We can give a basis of H, as follows. Let W be an index set for the G-orbits in X X, under the diagonal action of G. That is, X X = w W X w, where each X w is a G-orbit in X X. Let 1 be a distinguished element of W such that X 1 = {(x, x) : x X} is the diagonal orbit. We can think of X w as the set of pairs (x, x ) with a fixed difference, indexed by w. If we were to choose a basepoint x 0 X (which we will not), with stabilizer H < G, then W could be taken to be a set of representatives for the double cosets H\G/H. For each w W we define T w H by [T w f](x) = f(x ). x X (x,x ) O w In other words, T 1 = I is the identity endomorphism in H and in terms of the basis {e x } of V X we have T w e x = e x. x X (x,x ) O w Then {T w : w W } is an F -basis of H. One checks that for each g G. we have tr(ρ(g)t w, V X ) = {(x, x ) X w : x = g x}. (1) This is the number of x X whose difference from g x has index w. The algebra structure of H is closely related to the decomposition of V X into irreducible representations of G. To illustrate this, assume G is doubly-transitive on X. This means G is transitive on pairs (x, x ) with x x, so W = {1, w} has just two elements. The algebra H is quadratic, with basis {T 1, T w }. We have T 1 = I, and a single relation T 2 w = at 1 + bt w (2) for some scalars a, b F ; this relation completely determines the structure of H. We have (at 1 + bt w )e x = ae x + b x x e x. On the other hand, T 2 we x = y x e z, in which the coefficient of e x is a = X 1, and the coefficient of any e x, with x x, is b = X 2. Thus, the relation (2) may be written as z y (T w a)(t w + 1) = 0. 8

9 Now V X can be written as a direct sum of eigenspaces for T w : V X = V X (a) V X ( 1) for eigenvalues a and 1. From complete reducibility (Maschke s theorem, next section) it follows that both summands are irreducible representations of G. One checks that V X (a) consists of the constant functions and V X ( 1) = VX 0 consists of the functions whose sum of values is zero. Thus, when G is doubly transitive on X, the representation of G on VX 0 X 1. is irreducible, of dimension Examples: the reflection representations of symmetric and general linear groups The symmetric group S n+1 acts doubly-transitively on the set X = {0,..., n}. Hence we have an n-dimensional irreducible representation of S n+1 on the subspace R n := V 0 X = {f V X : f(i) = 0}. This is called the reflection representation of S n+1, because if F = R and we restrict the dot product on R n+1 to R n, then each transposition (i j) S n acts on R n via reflection about the hyperplane orthogonal to the vector e i e j R n. The group G = GL n+1 (q) acts doubly transitively on the set X = P n (q) of lines in F n+1 q. Hence we have an irreducible representation R n (q) of G, of dimension q + q q n. Even though G does not act by reflections on R n (q), this is called the reflection representation of G because, just as for S n+1, it corresponds to the character T w 1 of H = End G (V X ). This is one of many instances of GL n+1 (q) S n+1 as q 1 which some people regard as evidence for the field with one element. 2 Semisimplicity and characters In this chapter we assume G is a finite group whose order is nonzero in the field F. We will see that every finite-dimensional F -representation (ρ, V ) of G is a direct sum of irreducible representations and that the isomorphism class of V is completely determined by its character, which is the function χ ρ : G F given by χ ρ (g) = tr(ρ(g)). Assuming further that F is algebraically closed, we will prove that Irr F (G) is the number of conjugacy classes in G. 2.1 Projecting onto the invariants Recall that G 0 in F. In the group algebra F [G] we consider the element E := 1 G 9 g. g G

10 One easily verifies the properties This element E gives a linear operator ge = Eg = E for all g G, and E 2 = E. 1 P := ρ(e) = 1 G ρ(g) End(V ) g G for any finite-dimensional F -representation ρ : G GL(V ). From ge = Eg = E we conclude thatp End G (V ) and that P (v) = v for all v V G. From E 2 = E we have that P 2 = P. It follows that P (V ) = V G, that V = ker P V G, and that dim V G = tr(p ) = 1 G is the average value of the character χ ρ (g) = tr(ρ(g)). χ ρ (g) g G 2.2 Projecting onto the equivariants Now let (ρ, V ) and (ρ, V ) be two finite-dimensional F -representations of G. Recall that Hom G (V, V ) is the space of invariants in the representation ρ : G GL(Hom(V, V )), ρ (g)f = ρ f ρ 1. Applying the projection operator P for this representation ρ, we obtain the dimension formula dim Hom G (V, V ) = tr(p ) = 1 G χ ρ (g). To compute χ ρ (g) in terms of χ ρ and χ ρ we express Hom(V, V ) as a tensor product (over F ). First let ˇρ : G GL( ˇV ) be the representation of G on the dual space ˇV = Hom(V, F ), given by [ˇρ(g)λ](v) = λ(ρ(g) 1 v). g G Lemma 2.1 We have ˇV V Hom(V, V ), as F -representations of G. 1 An element in a ring squaring to itself is called an idempotent. 10

11 Proof: One checks that there is a well-defined F -linear map ˇV V Hom(V, V ), sending λ v to the endomorphism T λ v Hom(V, V ) given by T λ v (v) = λ(v)v. Next one checks that this map is G-equivariant. Finally, by choosing a basis in V, with dual basis in ˇV, one constructs the inverse of the above map, so it is an isomorphism of F -vector spaces, hence of F -representations. Next, one checks, again using a basis and its dual basis, that for all g G. It now follows that and we have proved the multiplicity formula: χˇρ (g) = χ ρ (g 1 ) χ ρ (g) = χ ρ (g 1 )χ ρ (g), dim Hom G (V, V ) = 1 G χ ρ (g 1 )χ ρ (g). (3) g G 2.3 Maschke s Theorem This is the fundamental theorem on representations of a finite group G whose order is nonzero in F. Theorem 2.2 (Maschke, 1899) Let ρ : G GL(V ) be a finite-dimensional F -representation of G, where G 0 in F. If U V is any G-invariant subspace, then there is another G-invariant subspace W such that V = U W. Proof: Take any basis of U and extend it to a basis of V. The basis vectors not in U span a subspace W 0 V such that V = U W 0. Let f 0 Hom(V, U) be the corresponding projection map, given by f 0 (u + w 0 ) = u. Let f = P (f 0 ), where P : Hom(V, U) Hom G (V, U) is projection onto the invariants in the representation Hom(V, U). Explicitly, we have f = 1 G If u U, we have each ρ(g) 1 (u) U, so f(u) = 1 G g G ρ(g) f 0 ρ(g) 1. g G ρ(g) f 0 ρ(g) 1 (u) = 1 G It follows that f Hom G (V, U) is surjective, and that V = ker f U. ρ(g) ρ(g) 1 (u) = u. g G Since f Hom G (V, U), the subspace W = ker f is a G-invariant complement to U in V. 11

12 Corollary 2.3 If G 0 in F and V is any finite dimensional F -representation of G then V = V 1 V n where each V i is an irreducible F -representation of G. To see that the condition G 0 in F is necessary, [ ] consider the representation ρ : Z/pZ GL 2 (p), 1 x given by consisting of matrices of the form with x F 0 1 p, acting in the natural way on V = F 2 p. 2.4 Schur s Lemma In this section there is no condition on G (it could be 0 in F or ), however we must assume F is algebraically closed. There is no harm in taking F = C, but we will not use properties particular to complex numbers. Theorem 2.4 (Schur s Lemma) Let V, V be two finite dimensional irreducible F -representations of G, where G 0 in the algebraically closed field F. Then End G (V ) = F I V consists only of scalar endomorphisms. And if V V then Hom G (V, V ) = 0. Proof: If f Hom = G(V, V ) is nonzero then ker f and im f are G-invariant subspaces of V and V respectively. The only possibility is that f is an isomorphism. Suppose this is the case. Since F is algebraically closed, f has an eigenvalue λ F. Then f λ I V End G (V ) has a nontrivial kernel, hence is identically zero. This means f is a scalar operator. Corollary 2.5 Let Z(G) be the center of G. If (ρ, V ) is an irreducible representation of G then there is a homomorphism ω : Z(G) F such that ρ(z) = ω(z) I V, for all z Z(G). The homomorphism ω is called the central character of (ρ, V ). Note that Schur s lemma is false if F is not algebraically closed. Consider G = SO(2) acting by rotations of the plane R Interwining and irreducibility Now assume that G 0 in F and that F is algebraically closed. Then Maschke s Theorem and Schur s Lemma both apply. We first have Theorem 2.6 (Strong Schur s Lemma) A finite-dimensional F -representation ρ : G GL(V ) is irreducible if and only if dim End G (V ) = 1. (The last condition means that End G (V ) consists only of scalar endomorphisms.) 12

13 Proof: If V is irreducible then dim End G (V ) = 1 by the old Schur s lemma. Conversely, suppose dim End G (V ) = 1 and that U is a nonzero G-invariant subspace of V. By Maschke, there is a G- invariant subspace W such that V = U W. The projection operator f : V U along W belongs to End G (V ) and is nonzero. Since f must be a nonzero scalar, its kernel W is zero. Hence U = V. 2.6 Multiplicity We continue to assume that G 0 in F and that F is algebraically closed. For convenience of notation, we define, for two F -representations (ρ, V ), (ρ, V ) of G, the intertwining number V, V = dim Hom G (V, V ). When it is necessary to specify the group G we will write V, V G. From equation (12), this pairing can be computed from the characters χ ρ and χ ρ by the formula V, V = 1 χ ρ (g 1 )χ ρ (g). G The pairing V, V is Symmetric: V, V = V, V. Bilinear: V, V V = V, V + V, V. g G The Strong Schur s Lemma says that V is irreducible if and only if V, V = 1. Hence the irreducibility of V can be determined from its character: we have 1 χ ρ (g 1 )χ ρ (g) 1, G with equality if and only if V is irreducible. g G In fact χ ρ determines the complete decomposition of V into irreducibles: If V = V 1 V n with each V i irreducible and W is an another irreducible F -representation of G, then V, W = {i : V i W } is the multiplicity of W in V. If Irr F (G) = {(ρ 1, W 1 ),..., (ρ h, W h )}, then V = V, W 1 W 1 V, W h W h, and the isomorphism class of V is completely determined by the integers V, W 1,..., V, W h. From (12) we have This proves V, W i = 1 G χ ρ (g 1 )χ ρi (g). g G 13

14 Proposition 2.7 If G 0 in the algebraically closed field F then two representations (ρ, V ) and (ρ, V ) of G are isomorphic if and only if their characters agree: χ ρ = χ ρ Isotypic components and multiplicity one The decomposition V = V 1 V n need not be canonical. For example if V = V 1 V 2 and V 1 V 2, then every isomorphism f : V 1 V 2 gives another G-invariant subspace V 3 = {(v, f(v)) v V 1 } and we also have V = V 1 V 3. However, if {W 1,..., W h } is a complete and list of irreducible representations of G, up to isomorphism and without repetition, then the subspace V Wi spanned by those V i which are W, is canonical, and the decomposition V = h i=1 is preserved by every intertwining operator T End G (V ). The space V Wi is called the isotypic component of V corresponding to W i. It is a sum of V, W i copies of the representation W i. In particular if V, W i = 1 then V Wi W i, so this summand is preserved by every endomorphism, and is canonical in this sense. We say that V is multiplicity-free if V, W i 1 for all i = 1,..., h. This too can be detected by endomorphisms. The proof of the following is now easy and left to the reader. Proposition 2.8 The representation V is multiplicity-free if and only if the algebra End G (V ) is commutative. V Wi 2.7 The Regular Representation Every finite group G has a canonical non-trivial representation, namely the permutation representation (ρ G, V G ) associated to the action of G on itself by left multiplication. Explicitly, the vector space V G has basis {e x : x G} and ρ G (g)e x = e gx. It is immediate that the character of the regular representation is given by { G if g = 1 χ ρg (g) = (4) 0 if g 1. Now if (ρ, V ) is any F -representation of G we have V G, V = 1 χ G (g 1 )χ ρ (g) = χ ρ (1) = dim V. G g G Thus, the decomposition of the regular representation into irreducibles is given by V G = dim(w ) W. W Irr F (G) 14

15 Since V G is finite dimensional, we see that Irr F (G) is finite and moreover, that G = V G, V G = (dim W ) 2. W Irr F (G) This formula is useful for proving that one has found all irreducible F -representations W of G. The fact that W appears in V G with multiplicity equal to its dimension is not a coincidence. For the larger group G G acts on G, via (g, h) x = gxh 1. One checks that the character is given by tr[(g, h), V G ] = C G (g) if g and h are conjugate in G, and tr[(g, h), V G ] = 0 otherwise. Using this it is now straightforward to prove following proposition. Proposition 2.9 The permutation representation of G G on G via (g, h) x = gxh 1 decomposes as V G = W ˇW. W Irr F (G) This is a special case of the Peter-Weyl Theorem for compact groups. 2.8 Permutation representations again Let H be a subgroup of G, so that G acts on X = G/H by left multiplication. The permutation representation (ρ X, V X ) has basis {e x : x X} where ρ X (g)e x = e gx. Let χ X be the character of ρ X. Now we have χ X (g) = {x X : gx = x} = X g, (5) the number of fixed-points of g in X. Proposition 2.10 If (ρ, V ) is any finite-dimensional F -representation of G then V X, V = dim V H. Proof: We compute: V X, V G = 1 G g G = 1 G = 1 G x X χ X (g 1 )χ ρ (g) = 1 G g G gx=x x X h H χ ρ (g) = 1 G χ ρ (h) = 1 H x X χ ρ (g) g G x X gx=x g G x χ ρ (g) χ ρ (h) = dim V H. h H 15

16 It follows that the transitive permutation representation V X on X = G/H decomposes as V X = dim(w H ) W. W Irr F (G) This time there is no right action of G on G/H. 2 However, the intertwining algebra H = End G (V X ) (see section 1.6.1) certainly acts on V X, so V X becomes a module over H F [G]. It can be shown that every simple module over H F [G] is of the form E W, where E Irr(H) and W Irr F (G). Now H is isomorphic to the subalgebra of elements α F [G] such that hαh = α for all h, h H. It follows that each W H is naturally an H-module. In fact W W H gives a bijection More details needed here. and {W Irr F (G) : W, V X 0} Irr(H), V X = W Irr F (G) ˇW H W is the decomposition of V X into simple H F [G]-modules. In the case of the Peter-Weyl theorem 2.9 we have H = 1 and the algebra H = F [G] for the right-acting copy of G. 2.9 The number of irreducible representations We continue to assume that F is an algebraically closed field in which G 0. Let ccl(g) denote the set of conjugacy-classes of G. A class function on G is a function f : G F which is constant on conjugacy-classes. That is, f(gxg 1 ) = f(x) for all g, x G. Regarding the group algebra F [G] as the space of all F -valued functions on G, we see that the class functions comprise the center of F [G]; we denote the space of class functions by z[g]. This is a commutative F -algebra with F -basis {f C : C ccl(g)}, so dim z(g) = ccl(g). Proposition 2.11 Irr F (G) = ccl(g). Proof: By Schur s lemma the functions χ ρ, for ρ Irr F (G) form an orthogonal set in z[g] with respect to the inner product on class functions given by It follows that Irr F (G) ccl(g). f, f = 1 G f(g 1 )f (g). g G To prove the reverse inequality, we let Y be the F -span of Irr F (G) and we will show that the map z[g] Hom(Y, F ) given by f f, is injective. 2 The normalizer N G (H) of H has a right action on G/H, but this does not explain the multiplicity dim V H. 16

17 For any f z[g] and irreducible represenation (ρ, V ), the endomorphism ρ(f) = g G f(g)ρ(g) End(V ) actually belongs to End G (V ), so by Schur s Lemma we have ρ(f) = ω ρ (f) I V, for some scalar ω ρ (f) F. Taking traces, we have ω ρ (f) dim V = f(g)χ ρ (g). g G If f, χ ρ = 0 for all ρ Irr F (G) then f, χˇρ = 0 for all ρ, which implies that ω ρ (f) = 0, so ρ(f) = 0 for all ρ Irr F (G). Now cor. 2.3 implies that ρ(f) = 0 for every finite dimensional F -representation (ρ, V ) of G. Applying this to the regular representation (ρ G, V G ), we have ρ G (f)e 1 = g G f(g)e g = 0, so f(g) = 0 for all g G so f = 0. Remark: Just as there is no canonical isomorphism between a vector space and its dual space, there is no canonical bijection between Irr F (G) and ccl(g). However, in certain cases there is a canonical mapping between Irr F (G) and conjugacy-classes in another group Ĝ, which is in some sense dual to G. For example, there is a canonical bijection between Irr F (S n ) and unipotent conjugacy classes in GL n (C), discovered by Springer in the 1970 s. 3 Induced Representations This section has the definition of an induced representation and the basic results: Frobenius reciprocity and Mackey s theorem, followed by short sections on Intertwining algebras and multiplicity-free representations. Again the field F is algebraically closed and G 0 in F. 3.1 Conjugate representations The group G acts on the set of its subgroups by conjugation: Given a subgroup H G and y G, set H y := y 1 Hy. More generally, G acts by conjugation on the set of pairs (H, U), where H is a subgroup of G and σ : H GL(U) is a representation of H. Namely, given y G, define σ y : H y GL(U) by σ y (h y ) = σ(h). We often denote this representation by U y, even though it occurs on the space U. The isomorphism class of U y depends only on the coset Hy. 17

18 3.2 Definition and first properties of induced representations Let G be a finite group, H G a subgroup. On the set X = H\G of left cosets, we define a left action of G by g x = g (Hy) = Hyg 1, for x = Hy. The stabilizer G x = {g G : g x = x} = H y. We define a representation G on the complex vector space of functions: Ind G H U = {f : G U : f(hg) = σ(h)f(g) h H, g G}, with G-action given by [g f](g ) = f(g g) for all g, g G. The support of any function in V is a union of left cosets of H. For x X, let V x be the set of functions in V supported on x. Then V = x X V x. (6) Under the action of g on V, each subspace V x is mapped to V g x. In particular G x preserves V x. This representation of G x is isomorphic to U y, where x = Hy. Indeed, we have a G x -isomorphism U y V x, u f u, where f u (hy) = ρ(h)u and f u 0 on every coset x x. In particular dim V x = dim U for all x X, so from (6) we have the dimension formula: dim Ind G H U = [G : H] dim U. The collection of pairs (G x, V x ) of groups and representations is compatible with the G-action, in the sense that for all g G and k G x, we have a commutative diagram k V x V x g V g x gkg 1 g V g x. It follows that the trace of g on V is given by We can write this more explicitly as tr(g, V ) = x X g x=x tr(g, V ) = Hy H\G Hyg=Hy tr(g, V x ). (7) χ σ (ygy 1 ), (8) where χ σ is the character of σ. Note that χ σ (ygy 1 ) is the same for all y x. The trace formula (8) shows that tr(g, V ) = 0 unless the G-conjugacy class of g meets H. 18

19 3.3 Induction and Restriction Given a representation η : G GL(W ) we can restrict η to a subgroup H G to obtain a representation (η H, W ) of H. We sometimes denote this representation by W H or Res G H W. Both Res G H and Ind G H are functors: Ind G H : Rep F (H) Rep F (G) and Res G H : Rep F (G) Rep F (H). In categorical terms, Frobenius Reciprocity says these two functors are adjoint. More precisely: Theorem 3.1 (Frobenius Reciprocity) Let σ : H GL(U) be a representation of H, let η : G GL(W ) be a representation of G. Then there is a unique isomorphism such that Φ(w)(1) = φ(w). Hom H (W H, U) Hom G (W, Ind G H U), φ Φ, Proof: Given Φ Hom G (W, Ind G H U)), define φ Hom H (W H, U) by φ(w) = Φ(w)(1). Given φ Hom H (W H, U), define Φ Hom G (W, Ind G H U)) by Φ(w)(g) = φ(gw). One checks that the correspondences φ Φ are mutual inverses. The numerical version of Frobenius reciprocity is the multiplicity formula W, Ind G H U G = W H, U H. Induced representations can be recognized as follows. Proposition 3.2 Let ρ : G GL(W ) be a representation of G. Suppose W is a direct sum of subspaces permuted transitively by ρ(g). Then W = W 1 W m W Ind G H W 1 where H = {h G : ρ(h)w 1 = W 1 } is the stabilizer of W 1 in G. 19

20 Proof: We note that dim W = [G : H] W 1 = dim V, so it suffices to find a G-equivariant surjection W V. The subspace W 2 W m is preserved by H, so there is an H-equivariant projection mapping φ : W W 1. By Frobenius reciprocity this gives G-equivariant mapping Φ : W V = Ind G H W 1, such that Φ(w 1 )(1) = φ(w 1 ) = w 1 for all w 1 W 1. If g / H, then gw 1 = W i for some i 1, so Φ(w 1 )(g) = φ(gw 1 ) = 0. Hence the image of Φ contains the subspace V 1 of functions in V supported on H. As the G-translates of V 1 span V, it follows that Φ is surjective, hence is an isomorphism. If K is another subgroup of G (possibly K = H), then for each orbit O of K in X the subspace V O := x O V x is preserved by K. Choose x O and y x. From Prop. 3.2, it follows that V O Ind K K x U y, where K x = K G x = K H y is the stabilizer of x in K. It follows that the restriction of V to K is a sum V K x X/K Ind K K x U y, (9) where the sum is over representatives of the orbits of K in X. Equation (9) is called Mackey s theorem. In the literature this is usually expressed as follows. Since X = H\G, the K-orbits in X are simply double cosets HyK. We have K x = K H y and V x = U y as representations of H y. Thus we have proved: Theorem 3.3 (Mackey s theorem) Let H, K be subgroups of G and let U be a representation of H. Then the restriction of Ind G H U to K is given by [ Ind G H U ] K Ind K K H U y. y HyK H\G/K Corollary 3.4 Let H, K be subgroups of G and let U, W be a representation of H and K respectively. Then Hom G (Ind G K W, Ind G H U) Hom K H y(w, U y ). HyK H\G/K Proof: This follows from Mackey s theorem and two applications of Frobenius reciprocity. Corollary 3.5 Assume that U is irreducible for H. Then the induced representation Ind G H U is irreducible for G if and only if Hom H H y(u, U y ) = 0 for all y G H. Corollary 3.6 If U = 1 H is the trivial representation of H then dim End G (Ind G H 1 H ) = H\G/H. 20

21 3.4 Representations of GL 2 (q) In this section we use our results so far to determine the irreducible representations of the group G = GL 2 (q), consisting of 2 2 invertible matrices over the field F q of cardinality q. 3.5 The Gelfand-Graev representation Take G = GL 2 (q), let N = representation [ ] 1 F q, and let ψ : N C be a nontrivial character of N. The Γ = Ind G N ψ was first studied by Gelfand and Graev in the early 1960s. Its isomorphism class is independent of the choice of nontrivial character ψ. We see immediately [ that ] dim[ Γ = [G ] : N] = (q 2 1)(q 1) and that the character χ Γ (g) = 0 unless g is conjugate to or to ([ ]) [ ] We compute χ Γ as follows. Let T be the diagonal matrices in G and let w =. Then [ ] a b g = lies in T N if c = 0 and in NwT N if c 0. In both cases g can be expressed uniquely in c d terms of the factors N, w, T, N. Hence T wt is a set of double coset representatives for N\G/N. [ ] 1 1 Take g =. If y wt then N N 0 1 y = I, so such y do not contribute to χ Γ (g). Let y T, and write y = (s, t) with t, s F q. Then so we have χ Γ (g) = (s,t) T ψ(s/t) = r ygy 1 = [ ] 1 s/t, 0 1 s/t=r ψ(r) = (q 1) r One checks that the number of conjugates of g is q 2 1. It follows that Γ, Γ = 1 [ χγ (1) 2 + (q 2 1)χ Γ (g) 2] = q(q 1). G ψ(r) = 1 q. This number is the sum of the squares of the multiplicities of the irreducible constitutents of Γ. In fact we will later see that Γ is multiplicity-free, hence has exactly q(q 1) distinct irreducible constituents. For example, the Steinberg representation of G is the subrepresentation on P 1 (q) consisting of functions whose sum of values is zero. (This is the reflection representation for n = 2, see section ) One checks that χ St (g) is the number of eigenlines of g, less one. Hence the supports of χ Γ and χ St intersect in the identity only. This means St, Γ = 1. 21

22 3.6 Principal series representations Keeping the notation above, let B = T N and let χ : T C be any character of T. This lifts to a character of B, trivial on N, via the projection B T. So χ is determined by two characters χ 1, χ 2 of F q. The representation I(χ) = Ind G B χ is part of the principal series, parameterized by χ. The conjugation of w on T switches χ 1 and χ 2. By abuse of notation we let χ w be the resulting lifted character of B (still trivial on N). We have I(χ) = I(χ w ), and one checks using Mackey that I(χ), I(χ ) 0 if and only if χ {χ, χ w. We note that the constituents V of principal series representations are exactly those for which V N 0. For V, I(χ) = V B, χ so if this is nonzero then V N 0 since χ is trivial on N. Conversely, if V N 0, then the space V N contains a character χ of T, so V, I(χ) 0. A principal series representation is much simpler to analyze than the Gelfand-Graev representation, because there are only two double cosets: G = B BwB, so I(χ) has at most two constituents. One checks that if χ 1 χ 2 then I(χ) is irreducible. If χ 1 = χ 2, then χ := χ 1 det is a character of G which restricts to χ. It follows that in this case we have I(χ) = χ ( χ St). Let us now compute the intertwining between I(χ) and Γ. We find that I(χ), Γ = χ, ψ y B N y = χ, ψ N + χ, ψ w B N w = 1, y N\G/B since χ ψ on N and B N w = {1}. Since in the case χ 1 = χ 2 we have χ ψ on N, it follows that χ St is the constituent of I(χ) appearing in Γ. Counting χ s modulo w, we have found that there are 1 (q 1)(q 2) + (q 1) = 1 q(q 1) 2 2 constituents of Γ coming from constituents of principal series representations, each appearing with multiplicity one. 3.7 Cuspidal representations A representation V of G is cuspidal if V N = 0. Such representations can only contain nontrivial characters of N, which are permuted transitively by T. As there are q 1 such characters, we find 22

23 that dim V = m V (q 1), for some integer m V representations. = V, Γ. In this section we will count the cuspidal First we count the conjugacy classes in G, by counting characteristic polynomials x 2 tx + d. As d 0 there are q(q 1) characteristic polynomials, each from a distinct conjugacy [ class, ] except [ those ] λ 0 λ 1 polynomials which are squares (x λ) 2, which come from either the class of or. It 0 λ 0 λ follows that ccl(g) = 2(q 1) + [(q(q 1) (q 1)] = q 2 1. By our analysis above, the number of constituents of principal series representations is 2(q 1) + 1 (q 1)(q 2). 2 Subtracting this from q 2 1, it follows that there are 1 q(q 1) cuspidal representations. As Γ, Γ = 2 q(q 1), we find that each cuspidal representation appears in Γ with multiplicity one, hence has dimension q 1. We also see that Γ is multiplicity free, consisting precisely of the representations of G of dimension > 1. 4 Representations of S n Let F be an algebraically closed field in which n! 0. In this section we construct some families of irreducible F -representations of the symmetric group S n and then sketch the classification of all irreducible representations of S n. 4.1 Exterior power representations We know three irreducible representations of S n already: 1 n = 1 Sn = trivial rep., R n = reflection rep. sgn n = sign character. Recall that R n is realized on the hyperplane in F n given by x x n = 0. These are part of the family of exterior power representations: Proposition 4.1 For 0 k n 1 the exterior powers Λ k (R n ) are irreducible representations of S n. Proof: We argue by induction on n. Regard S n 1 S n as the stabilizer of n. Let U = {(x 1,..., x n 1, 0) F n : i x i = 0} and let v = (1, 1,..., 1, 1 n) R n. Then U = R n 1 is the reflection representation of S n 1 and R n = R n 1 F v. It follows that Λ k (R n ) = Λ k (R n 1 ) v Λ k 1 (R n 1 ). 23

24 By induction these two summands are irreducible for S n 1. And they are non-isomorphic because their dimensions are different. Hence they are the only proper invariant subspaces for S n 1 in Λ k (R n ). Now R n has the basis {α 1,..., α n 1 } where α i = e i e i+1, and {α 1,..., α n 2 } is a basis of R n 1. The transposition (n 1 n) sends α n 2 α n 2 + α n 1 hence does not preserve Λ k (R n 1 ). It follows that Λ k (R n ) has no proper S n -invariant subspaces. Exercise: Show that Λ k (F n ) = Ind Sn S k S n k (ε k 1 n k ) = k Λ i (R n ). i=0 4.2 Grassmannian representations Let X k be the set of k-element subsets of [1, n] = {1,..., n}. This set is to S n as the Grassmannian of k-dimensional subspaces of F n q is to GL n (q). Let H k S k S n k be the stabilizer of the point [1, k] X k, so the permutation representation of S n on X k is V k := V Xk = Ind Sn H k 1 Hk. If j k, the partition of X k into H j -orbits is given by X k = k X k (p), where X k (p) = {A X k : A [1, j] = p}. From Cor. 3.6, it follows that p=0 V j, V k = j + 1 for all j k. Proposition 4.2 For 0 k n/2 there are distinct irreducible representations Γ k of S n uniquely characterized by the conditions { 1 if j k V j, Γ k = 0 if j < k. Proof: This is clear if k = 0, for V 0 = Γ 0 = 1 n and V j, Γ 0 = 1 for all j 0. Assume Γ 0,..., Γ m have been found. Then V m+1, Γ k = 1 for all k m, so V m+1 contains Γ 0,..., Γ m. As V m+1, V m+1 = m + 2, there can be only one more irreducible constituent of V m+1, which must be distinct from Γ 0,..., Γ m. This is Γ m+1. It follows that the induced representation V k = Ind Sn H k 1 Hk decomposes as V k = Γ 0 Γ k for k n/2 and that dim Γ k = ( ) ( ) n n. (10) k k 1 24

25 4.3 General irreducible representations of S n The number of irreducible representations of S n is the number of conjugacy classes. The latter are parameterized by partitions λ = [λ 1, λ 2,..., λ n ] of n, where λ 1 λ 2 λ p > 0 and λ i = n. The number p = p(λ) is the number of (nonzero) parts of λ. For general groups there is no canonical bijection between irreducible representations and conjugacy classes, however for S n we can also associate an irreducible representation ρ λ : S n GL(V λ ) to each partition λ of n. We start by letting S λ be the subgroup preserving each subset {1, 2,..., λ 1 } {λ 1 + 1,..., λ 1 + λ 2 }... {λ λ n 1 + 1,..., λ λ n 1 + λ n }. So S λ S λ1 S λt. Now let 1 λ be the trivial representation fo S λ. The induced representation I λ = Ind Sn S λ 1 λ is the permutation representation of S n acting on the set X λ whose elements are set partitions where A λi = λ i and p = p(λ). Define a partial order on partitions by: The upper and lower regions look like {1, 2,..., n} = A λ1 A λ2 A λp λ µ if λ 1 µ 1, λ 1 + λ 2 µ 1 + µ 2, etc. [n] [n 1, 1] [n 2, 2] [n 2, 1, 1] [3, 1 n 3 ] [2, 2, 1 n 4 ] [2, 1 n 2 ] [1 n ]. It is not a total ordering: [3, 1, 1, 1] and [2, 2, 2] are not comparable. Next, to λ = [λ 1, λ 2,... ] we associate a Young diagram. This is a left-justified array of λ 1 boxes on λ 2 boxes, etc., and in each box b we put the number of boxes directly below and to the right of b, including b itself. These numbers are called the hooklengths of λ. For example: λ = [5, 2, 2, 1] hook lengths We state without proof the following Theorem 4.3 For each partition λ of n there is an irreducible representation V λ of S n which is uniquely characterized by the property I λ = V λ µ>λ m λµ V µ, 25

26 with m λµ non-negative integers, and we have where the h i are the hooklengths of λ. dim V λ = n! h 1 h 2 h n, In other words, V λ appears with multiplicity one in I λ and all other constituents of I λ are of the form V µ with µ > λ. In the example above with λ = [5, 2, 2, 1] we would have dim(v λ ) = (10)! = 525. If λ = [n k, k] with n k k then the only partitions µ λ are of the form [n j, j], with 0 j k. In the notation of section 4.3 we have so I λ = Γ Γ k, W [n k,k] = Γ k is the Grassmannian representation from section 4.3. The Young diagrams of the partitions [n k, k] are precisely those with two rows. For example the diagram of λ = [5, 3] with its hooklengths is and the hooklength formula gives in agreement with (10) above. dim V [5,3] = , 8! = 28 = ( ) 8 3 ( ) 8, 4 The exterior power representation Λ k (R n ) has partition The partitions λ = [n k, 1 k ] are called hook partitions because of the shape of their Young diagrams, e.g. for [5, 1, 1, 1]: is Hook partitions correspond to the exterior power representations: Λ k (R n ) = V [n k,1 k ]. Note that the hooklength formula gives ( ) n 1 dim(λ k (R n )) =, k as it should. 26.

27 4.4 The representations of S 5 The representations of S 5 are all constructed from exterior power or Grassmanian representations. We list the partitions of 5 in descending order (which is linear for n = 5), along with some decompositions of I λ and the corresponding irreducible representations V λ. λ I λ V λ dim V λ [5] [4, 1] 1 5 R 5 R 5 4 [3, 2] 1 5 R 5 Γ 2 Γ 2 5 [3, 1, 1] 1 5 2R 5 Γ 2 Λ 2 (R 5 ) Λ 2 (R 5 ) 6 [2, 2, 1] 1 5 2R 5 (sgn 5 Γ 2 ) sgn 5 Γ 2 5 [2, 1 3 ] 1 5 3R 5 Λ 3 (R 5 ) Λ 3 (R 5 ) 4 [1 5 ] 1 5 4R 5 sgn 5 sgn More on intertwining algebras Let ρ : G GL(V ) be any finite dimensional complex representation of G and let H = End G (V ) be the algebra of G-equivariant endomorphisms of V, under composition. For each irreducible representation W of G, let W be the isotypic component of W in V. The action of H on V preserves W and commutes with the G action. Hence W is a module over the tensor product algebra H C[G]. From complete reducibility it follows that W is a simple H C[G]-module. From [CR 10.38(iii)] it follows that W is a tensor product W E W W, where E W is a simple H-module. Now Thus, Hom G (W, V ) = Hom G (W, W ) = Hom G (W, E W W ) = E W Hom G (W, W ) = E W. E W = Hom G (W, V ), where ϕ H acts via [ϕ(f)](w) = ϕ(f(w)), for w W and ϕ Hom G (W, V ). Now H = End G (V ) = W End G ( W ) = W End G (E W W ) = W End(E W ), (11) where the sums are over the irreducible constituents for G in V. It follows that the the correspondence W E W gives a bijection between the set of irreducible G-constituents W of V and the set of simple H-modules, such that dim E W = dim Hom G (W, V ), (12) 27

28 and as G-representations we have V = W dim(e W ) W. Now suppose H G and ρ : H GL(U) is a representation of H. Let V = Ind G H U, as in the previous section. Cor. 3.4 gives an isomorphism of vector spaces End G (V ) = Hom H H x(u, U x ). (13) x H\G/H To see this as an algebra isomorphism we use a different realization of End G (V ), as follows Let H(G, H, ρ) = {ϕ : G U : ϕ(hgk) = ρ(h)ϕ(g)ρ(k), g G, and h, k H}. This is an algebra under convolution product which can be expressed in several ways: ϕ 1 ϕ 2 (g) = 1 H (x,y) G G xy=g ϕ 1 (x)ϕ 2 (y) = 1 H ϕ 1 (x)ϕ 2 (x 1 g) = x G x G/H ϕ 1 (x)ϕ 2 (x 1 g). This product is associative but not necessarily commutative. We call H(G, H, ρ) the Hecke algebra of the pair (H, ρ). The Hecke algebra acts on V = Ind G H U via convolution, as (ϕ f)(g) = 1 H ϕ(x)f(x 1 g), (14) for ϕ H(G, H, ρ) and f V. One checks that (ϕ ψ) f = ϕ (ψ f), so that we have an algebra homomorphism x G H(G, H, ρ) End G (V ), ϕ ϕ ( ). (15) If f V is supported on Hx and f(x) = u U, one checks that ϕ f(1) = ϕ(x 1 )u. It follows that the homomorphism (15) is injective. Each function ϕ H(G, H, ρ) is supported on a union of H H double cosets in G. The functions supported on a single double coset HxH form a linear subspace of H(G, H, ρ) isomorphic to Hom H H x(u, U x ) via the map ϕ ϕ(x). From (13), it follows that H(G, H, ρ) and End G (V ) have the same dimension, so (15) is in fact an isomorphism of algebras. 5.1 Multiplicity-free representations We say that a representation V of G is multiplicity-free if dim Hom G (W, V ) 1 for all irreducible representations W of G. In other words, V is multiplicity-free when each isotypic component of V is irreducible. This means that each irreducible constituent of V has a canonical realization in V. From (11) we have the following Corollary 5.1 The representation V is multiplicity-free if and only if the interwining algebra End G (V ) is commutative. 28

29 If V = Ind G H ρ, one can sometimes prove that End G (V ) is commutative by using the following trick. Lemma 5.2 Suppose there exists a map α : G G with the following properties: 1. α(xy) = α(y)α(x) for all x, y G; 2. α(h) = H; 3. If Hom H H x(ρ, ρ x ) 0 then there exists x HxH such that α(x) = x. Then H(G, H, ρ) is commutative and Ind G H ρ is multiplicity-free. Proof: Given α as above, define a map α : H(G, H, ρ) H(G, H, ρ) by One checks that α(ϕ) = ϕ α. α(ϕ 1 ϕ 2 ) = α(ϕ 2 ) α(ϕ 1 ). Condition 2 implies that α permutes the H H double cosets in G. Suppose ϕ is supported on a single H H double coset. Condition 3 implies that this double coset is of the form HxH with α(x) = x. We then have α(hxh) = HxH and α(ϕ) is also supported on HxH. Since α(ϕ)(x) = ϕ(α(x)) = ϕ(x), we have α(ϕ) = ϕ, so α is the identity map on H(G, H, ρ). Hence for ϕ 1, ϕ 2 in H(G, H, ρ) we have ϕ 1 ϕ 2 = α(ϕ 1 ϕ 2 ) = α(ϕ 2 ) α(ϕ 1 ) = ϕ 2 ϕ 1, so H(G, H, ρ) is commutative. It now follows from Cor. 5.1 that Ind G H ρ is multiplicity-free. 6 Representations of GL 2 (q), again. In this section we construct the representations of the group G = GL 2 (q), using the Gelfand-Graev representation. The emphasis is on understanding representations directly, minimizing the use of characters. The Gelfand-Graev representation is the induced representation where N = Γ := Ind G N ψ, {[ ] } 1 c : c F 0 1 q F + q and ψ : F + q C is a nontrivial character of N. If ψ is another nontrivial character of N then there is a diagonal matrix a G such that ψ = ψ a. It follows that the isomorphism class of Γ is independent of the choice of nontrivial character ψ. This is another version of earlier treatment. Can t decide which to keep. 29

30 Proposition 6.1 The Gelfand-Graev representation Γ is multiplicity-free and is the direct sum of all the irreducible representations V of G for which dim(v ) > 1. Proof: From the Bruhat decomposition, it follows that the N N double coset decomposition of G is G = NA NAwN, [ ] [ ] 0 1 a 0 where A is the subgroup of diagonal matrices, and w =. Suppose x = A d Then N N x = N and if a d we have ψ x ψ, so Hom N (ψ, ψ x ) = 0, while if a = d then Hom N (ψ, ψ x ) = End N (ψ) is one-dimensional. Next, if x Aw, then N N x = {I}, so Hom N N x(ψ, ψ x ) is automatically one-dimensional. Thus, we find that the Hecke algebra H(G, N, ψ) is supported on Z Aw, where Z is the center of G. One checks that the map α : G G defined by ([ ]) [ ] a b d b α = c d c a satisfies each of the conditions of lemma 5.2, so Γ is multiplicity-free. Now let ρ : G GL(V ) be an irreducible representation of G. If dim V = 1 then V is trivial on the commutator subgroup [G, G] = SL 2 (q), which contains N, hence V cannot contain ψ and Hom G (V, Γ) = 0. Conversely, suppose V does not contain ψ. Since A is transitive on nontrivial characaters of N it follows that V is trivial on N. Then V is also trivial on the conjugate subgroup [ ] 1 0 N = wnw 1 =. 1 Since N and N generate SL 2 (q), it follows that ρ factors through SL 2 (q), hence is one dimensional. Therefore the irreducible representations V appearing in Γ are precisely those of dimension greater than one. The irreducible representations V of G are of two kinds, depending on whether V N = 0 or not. Proposition 6.2 The irreducible representations V of G with V N 0 are as follows. 1. Ind G B χ where χ : B C is trivial on N; 2. χ det, where χ : F q C is a character. 3. St G χ det, where St G is the unique nontrivial constituent of Ind G B 1 B, of dimension q. Proof: Exercise using Frobenius reciprocity and Mackey s theorem. The characters χ det are trivial on N hence do not appear in Γ. From Prop. 6.2 it follows that if V is an irreducible representation of G appearing in Γ and V N 0, then dim(v ) = q + 1 or dim(v ) = q in cases 1 and 3, respectively. If V N = 0 we say that V is cuspidal. From Prop. 6.1 we deduce the following 30

31 Corollary 6.3 If V is an [ irreducible ] cuspidal representation of GL 2 (q) then dim V = q 1 and V M Ψ, where M = and Ψ = Ind M N ψ is the unique irreducible representation of M 0 1 containing a non-trivial character of M (see exercise above). In particular all cuspidal representations of G remain irreducible and become isomorphic to each other when restricted to M. Proof: Since V is cuspidal, it contains only nontrivial characters of N. Since ψ, V N = Γ, V G = 1, each nontrivial character ψ appears exactly once in V. Since there are q 1 nontrivial characters of N, we have dim V = q 1. Since Ψ, V M = ψ, V N = 1 and dim Ψ = q 1 it follows that V M Ψ. We will find all the cuspidal representations as follows. For simplicity we assume q is odd. Let ɛ F q be a non-square and consider the subgroup 3 {[ ] a bɛ S = b a } : a or b 0. [ ] [ ] [ ] 1 [ ] a b ɛ 0 ɛ 0 a cɛ For g = G, we define σ(g) = g c d 0 1 T =. Then σ(s) = s for all 0 1 b/ɛ d s S. I claim that if g G S then there exists s [ S such ] that σ(sg) = sg. We can choose x ɛy x, y F q to satisfy (b cɛ)x = ɛ(a d)y, and then s = works. y x It follows that Ind G S η is multiplicity-free, for any character η of S. One checks that the following are equivalent: 1. η factors through the norm F q 2 F q. 2. η extends to a character of G. 3. η = η q. Say that η is regular if it does not satisfy 1-3. Then for all χ, so Ind G S η is a subrepresentation of Γ. χ det, Ind G S η G = 0 We can refine this: Fix a character ω of Z and suppose η is a regular character of S such that η Z = ω. Then Ind G S η is contained in Γ ω := Ind G NZ ψ ω, 3 If we regard the quadratic extension f = F q ( ɛ) as a two dimensional vector space over F q, then S is the isomorphic image of f acting on f by multiplication. Using the basis {1, ɛ}, the matrix above corresponds to a + b ɛ. 31