The Weil Representation Charlotte Chan June 7, 2012 Abstract This is my senior honors thesis done in my final year as an undergraduate at Stanford University, under the direction of Professor Akshay Venkatesh. We will construct the Weil representation of SL2(R) through a natural action of the Heisenberg group Heis(R) on the space of square-integrable 2 complex-valued functions L (R), together with the celebrated Stone-von Neumann theorem of functional analysis. Our approach will be to explicitly construct the analogous representation of 2 SL2(Fp ) on L (Fq ), for p an odd prime, using the finite-field equivalents of the aforementioned ingredients. This will allow us to separate the functional analytic complications of unitary representation theory of Lie groups from the representation theoretic and purely algebraic ideas. As the final arc to our triptych of stories, we will construct the Lie algebra analogue of the Weil representation through the Heisenberg algebra. Throughout this paper, we will be as explicit as possible while simultaneously giving motivation to our computations, allowing us to speak of concepts concretely without sacrificing the overarching philosophy. In spite of these efforts to make this paper self-contained, some proofs are omitted to avoid straying from the main arc of the story of the Weil representation. We will draw upon ideas from homological algebra, group theory, algebraic topology, functional analysis, measure theory, Lie theory, and, of course, representation theory. What is remarkable is that this topic touches as many subject areas as this paper will use, including number theory, harmonic analysis, topology, and physics. Because of its place in the intersection of a multitude of different fields, the Weil representation is also known, in the literature, as the Segal-Shale Weil representation, the metaplectic representation, the harmonic representation, and the oscillator representation. Contents 1 Introduction3 2 The Heisenberg Group7 2.1 The Structure of the Heisenberg Group as a Central Extension............7 2.2 The Action of SL2(K) .......................................8 2.3 The Heisenberg Representation................................ 12 3 The Finite-Field Model 14 3.1 The Representation Theory of Heis(Fp ) ........................... 14 3.2 Constructing a Map SL2(Fp ) PGLp (C) .......................... 15 3.3 Lifting Projective Representations! ............................... 20 3.4 The Schur Multiplier of SL2(Fp ) ................................ 23 3.5 The Weil Representation of SL2(Fp ) .............................. 25 4 The Real Case 26 4.1 The Representation Theory of Heis(R) ........................... 26 2 4.2 Constructing a Map SL2(R) PGL(L (R)) ........................ 27 ! 4.3 The Weil Representation for SLf2(R) ............................. 30 5 The Lie Algebra sl2(R) 32 5.1 The Lie Derivative......................................... 32 5.2 The Heisenberg Algebra..................................... 34 5.3 The Irreducibility of the Heisenberg Algebra Representation............. 36 5.3.1 Hermite Functions................................... 37 5.3.2 The Proof of Irreducibility.............................. 39 5.4 The Lie Algebra Analogue.................................... 40 5.5 Defining the Weil Representation for sl(2) ......................... 44 5.6 The Module Structure of the Weil Representation.................... 46 Acknowledgements 50 References and Bibliography 51 Colophon 53 1 Notation and Conventions K, a field G, a group g, a Lie algebra n 1 a b o Heis(K) := 0 1 c : a, b, c K , the Heisenberg group of K 0 0 1 2 Z(G), the center of G , an irreducible character of Z(Heis(K)) 2 L (K), square-integrable C-valued functions on K (see further comments on page 13) ¦ a b © SL2(K) := : a, b, c, d K, ad b c = 1 c d 2 − GL(V ), the space of invertible K-linear transformations V V for a vector space V over K ! End(V ), the space of K-linear transformations V V for a vector space V over K ! (R), the Schwartz space, the space of rapidly decreasing functions on R S ¦ a b © sl2(K) := : a, b, c, d K, a + d = 0 , the Lie algebra of SL2(K) c d 2 sl(2) := sl2(R) C, the complexified Lie algebra of SL2(R) ⊗R n 0 a c o heis(K) := 0 0 b : a, b, c K , the Heisenberg algebra of K, the Lie algebra of the 0 0 0 2 Heisenberg group Heis(K) heis := heis(R) C, the complexified Lie algebra of Heis(K) ⊗R Isom(H ) := φ : H H : φ(v),φ(w) = v, w , where , is the inner product with f ! h i h ig 〈· ·〉 which H is endowed. R+, the real numbers viewed as an additive group Vλ, a lowest weight module of weight λ Unless otherwise stated, all representations will be complex representations. 2 1 Introduction In this paper, we give three parallel constructions of Weil representation through the representa- tion theory of the Heisenberg group (or, in the Lie algebra case, the Heisenberg algebra). Namely, we construct the Weil representation for SL , SL , and for sl 2 : sl . In each 2(Fp ) 2(R) ( ) = 2(R) R C of these cases, we begin with a Heisenberg object (Heis , Heis , and heis : ⊗heis , (Fp ) (R) = (R) R C respectively) and study its representation theory. From this, we will construct a projective⊗ repre- sentation (of SL2(Fp ), SL2(R), and sl(2), respectively), and then lift this projective representation to a linear representation. This resulting representation is exactly the Weil representation and has a surprisingly significant role in many fields of mathematics, including number theory, topology, harmonic analysis, and physics. Unless otherwise stated, all representations in our discussion are complex representations. We will begin by discussing the Heisenberg group 80 1 9 1 a c <B C = Heis(K) = B0 1 bC : a, b, c K @ A 2 : 0 0 1 ; for a field K with characteristic away from 2. Section2 is completely focused on this discussion. 2 2 We will explicitly analyze its structure as a central extension of K⊕ by K, where K⊕ and K are viewed as additive groups. Hence we have a short exact sequence 2 0 K Heis(K) K⊕ 0. (1.1) ! ! ! ! Writing Z Z Heis K and Q Heis K Z, we obtain isomorphisms Z K and Q K 2. = ( ( )) = ( )= ∼= ∼= ⊕ Letting SL K act by matrix multiplication on Q K 2 and trivially on Z K, we obtain an 2( ) ∼= ⊕ ∼= action of SL2(K) on Heis(K). This is described explicitly by unravelling the second cohomology 2 class in H (Q,Z) associated to the isomorphism class of central extensions described by (1.1). This action of SL2(K) on Heis(K) is the first main ingredient in our construction of the Weil representation. The second main ingredient is the following Heisenberg representation associated to a central character. For an irreducible character : Z(Heis(K)) C, we will define a representation π on 2 ! 2 the C-vector space L (K) by the following action on f (x) L (K): 2 0 1 1 a c B C B0 1 bC f (x) := ( b x + c)f (x a). @ A · − − 0 0 1 In this paper, we will only discuss the specializations to K = Fp and K = R. In the former case, 2 L (K) has the obvious meaning as a p-dimensional C-vector space of C-valued functions on K, 3 2 and in the latter case, L (K) is defined to be the C-vector space of square-integrable functions on K with respect to the Lebesgue measure. This Heisenberg representation is discussed in Section 2.3. Given these two ingredients, the natural subsequent question is, “What happens when we put them together?” That is, what is the relationship between the representation π and the representation π g obtained by precomposing with the action of g SL2(K)? This is where the story ramifies into◦ the finite-field case and the real case. 2 The (complex) representation theory of Heis(Fp ) is simple. From the representation theory of finite groups, we know that the dimension of an irreducible representation of some finite group G must divide the order of G. Furthermore, the decomposition of the regular representation gives us that X 2 dim(V ) = G , j j where V ranges over a transversal of the isomorphism classes of irreducible representations of G. 3 2 Since Heis(Fp ) = p , we obtain arithmetically that Heis(Fp ) has p irreducible representations of dimensionj 1jand p 1 irreducible representations of dimension p. We in fact can say more − about the representations of Heis(Fp ): Theorem. For a non-trivial irreducible character :Z(Heis(Fp )) C, there exists a unique, up to ! isomorphism, representation of Heis(Fp ) such that Z(Heis(Fp )) acts by . This is the finite-field analogue of the celebrated Stone-von Neumann theorem from functional analysis. 2 It follows from the above theorem that the representation (π ,L (Fp )) is irreducible and that 2 this representation is isomorphic to (π g,L (Fp )) for any g SL2(Fp ). This means that there ◦ 2 2 exists an intertwining operator Φg : Heis(Fp ) GL(L (Fp )) = GLp (C) such that ! Φg (h)π (h) = (π g)(h)Φg (h), for all h H. ◦ 2 By Schur’s lemma, an intertwining operator between isomorphic irreducible representations is uniquely determined up to a scalar. Hence, from the Heisenberg representation together with the action of SL2(Fp ) on Heis(Fp ), we get the projective representation SL2(Fp ) PGLp (C), g [Φg ]. ! 7! (The notation [Φg ] means the image of Φg with respect to the surjection GLp (C) PGLp (C).) In Section 3.2, we explicitly compute this projective representation. ! We would like to lift this projective representation to a linear representation. In Section 3.3, we discuss the problem of lifting projective representations of a group G in general.