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Introduction to the Theta Correspondence

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Introduction to the Theta Correspondence Do you even lift? Introduction to the theta correspondence Kevin Chang June 15, 2021 Contents 1 Introduction 1 1.1 Notation and conventions . .2 2 The Weil representation and the theta correspondence 3 2.1 The Weil representation over a p-adic field . .3 2.2 The Weil representation over a finite field . .5 2.3 Main example: The theta correspondence for (Sp(2); O+(4)) ...6 3 The preservation principle 7 3.1 The preservation principle for the local theta correspondence . .7 3.2 Main example: A chain of supercuspidals . .8 3.3 The preservation principle over a finite field . .9 3.4 Main example: The preservation principle for (Sp(2); O+(4)) ..9 4 Compatibility 10 4.1 A result of Pan . 10 + 4.2 Main example: Compatibility between p-adic (Sp(V2); O(V6 )) and finite (SL(2); O−(2) × O+(4)) .................. 11 4.2.1 Components are not cuspidal . 11 4.2.2 Components are cuspidal . 13 5 Continuation of the main example 13 1 Introduction As every Hollywood executive knows, if there’s ever a good thing, it needs to be milked dry. The local theta correspondence is a powerful method for pro- ducing representations of p-adic groups following precisely this philosophy. In slightly less vague terms, the local theta correspondence is a matching between certain representations of certain pairs of p-adic reductive groups known as dual reductive pairs. If we have a chain of p-adic groups with each consecutive pair a 1 dual reductive pair and a representation of the first group in the chain, we can try to produce a chain of representations by repeatedly applying the local theta correspondence. The local theta correspondence is especially useful for the study of super- cuspidal representations, which are especially difficult to construct. The preser- vation principle of Howe and Kudla is a recipe for picking our chain of p-adic reductive groups such that two miracles occur: (1) the chain of representations produced by the local theta correspondence can be extended forever (2) every representation in it is supercuspidal. There is an analogue of the local theta correspondence for dual reductive pairs over finite fields. We call this the finite theta correspondence. Although the finite theta correspondence is not as nice as the local theta correspondence, it enjoys a variant of the preservation principle due to Pan. Another powerful tool for producing representations of finite reductive groups is Deligne-Lusztig induction, and as may be expected, the finite theta correspondence and Deligne- Lusztig induction are related in nice ways. This paper is an exposition of several different topics related to the theta correspondence, covering both the local theta correspondence and the finite theta corresopndence. For concreteness, we also work out several relatively well-understood examples, which recur throughout this paper. In Section 2, we review the construction of the Weil representation and introduce the theta correspondence. We also state a formula of Srinivasan and Pan that explicitly decomposes the uniform part of the Weil representation for certain finite dual reductive pairs in terms of Deligne-Lusztig virtual characters. In Section 3, we state the preservation principle and explain its use for constructing chains of supercuspials and cuspidals. In Section 4, we discuss a result of Pan that establishes some compatibility between the local theta correspondence and the finite theta correspondence. In Section 5, we discuss some proposed steps for extending our methods from the well-understood examples to an entire chain of groups. 1.1 Notation and conventions Throughout, we use F to denote a fixed p-adic field with p odd, we use f to denote its residue field, and we use $ to denote a fixed uniformizer of F . We use 1 to denote the trivial representation and sgn to denote the sign representation for orthogonal groups. If V (resp. v) is an F -vector space (resp. f-vector space) with a bilinear form, we let G(V ) denote the subgroup of GL(V ) (resp. GL(v)) that preserves the bilinear form. We will mainly use G(V ) and G(v) to denote orthogonal and symplectic groups. Any special orthogonal group SO(V ) (over any field of odd characteristic) has a unique automorphism given by conjugation by any element of of its com- plement in O(V ). We will denote this automorphism σ throughout this paper. 2 2 The Weil representation and the theta corre- spondence We briefly review the Weil representation and the theta correspondence, both in the p-adic field and finite field setting. 2.1 The Weil representation over a p-adic field The primary reference for this section is the first two chapters of [4]. We first review the construction of the Weil representation. Let be an additive character of F , and let (W; h; i) be a symplectic F -vector space. The Heisenberg group H(W ) := W ⊕ F has multiplication 1 (w ; t )(w ; t ) = w + w ; t + t + hw ; w i : 1 1 2 2 1 2 1 2 2 1 2 By the Stone-von Neumann, there exists a unique smooth irreducible repre- sentation (ρ ;S) of H(W ) with central character , i.e. F ⊂ H(W ) acts by . Note that Sp(W ) has a right action on H(W ): (w; t)g = (g−1w; t): g We can twist (ρ ;S) by this action: for any element g 2 Sp(W ), let (ρ ;S) be g defined by ρ (h) = ρ (hg). Since the action of Sp(W ) on the center F ⊂ H(W ) g is trivial, ρ is a smooth irreducible representation of H(W ) with central char- acter . By the uniqueness of (ρ ;S), there is an automorphism A(g): S ! S −1 g such that A(g) ρ (h)A(g) = ρ (h). This automorphism is unique up to scal- ing by an element of C×, since by Schur’s lemma, the only automorphisms of (ρ );S) are given by scalar multiplication. Thus, we have a projective repre- sentation of Sp(W ) on S. The projective representation of Sp(W ) becomes a bona fide representation when we pull it back to a double cover Sp(^W ) ! Sp(W ) called the meta- plectic group. We refer to this representation as the Weil representation (determined by the additive character ). Having a representation of Sp(^W ) is useful for pairing representations of dual reductive pairs, which are pairs of reductive subgroups of Sp(W ) that are mutual centralizers. More precisely, let V and V 0 be F -vector spaces equipped with bilinear forms. Then a pair of subgroups G; G0 ⊂ Sp(W ) is a dual reductive 0 0 0 0 pair if G and G are reductive, G = CSp(W )(G ), and G = CSp(W )(G). If (G; G ) is a dual reductive pair, then their preimages G;e Gf0 ⊂ Sp(^W ) centralize each 0 other, so we can pull back ! along the map Ge × Gf ! Sp(^W ). For simplicity (and due to the author’s lack of knowledge), for many of the later results, we will restrict to dual reductive pairs of the form (Sp(V ); O(V 0)), 3 where V is a symplectic F -vector space, V 0 is an even-dimensional quadratic F - vector space, and V ⊗ V 0 = W . Other good examples for which similar results (to the ones we will describe later) hold include pairs of unitary groups and symplectic/odd-orthogonal pairs (these are called pairs of type I). One reason that symplectic/even orthogonal pairs are convenient is that the maps Sp(^V ) ! Sp(V ) and O(^V 0) ! O(V 0) admit sections. Thus, once we pick sections, the Weil 0 representation ! pulls back to a representation of Sp(V ) × O(V ). Sometimes, the choice of sections matters; we will say if this is the case. When it does not, we will implicitly fix choices of sections for every symplectic/even orthogonal pair throughout. Of course, there are dual reductive pairs that are not symplectic/even or- thogonal, but there does exist a classification of p-adic dual reductive pairs (see [8, Chapter 1]). The irreducible dual reductive pairs are as follows (up to switching the order of the groups): (1) (U(V ); U−(V 0)): Let D be either F , a quadratic extension of F , or a division quaternion algebra over F . Fix = ±1. Here, V is an -skew Hermitian right D-vector space, and V 0 is a −-skew Hermitian left D- vector space. 0 (2) (GLD(V ); GLD(V )): Here, D is a central division algebra over F , and V and V 0 are right vector spaces over D. Pairs of types (a), such as symplectic/orthogonal pairs (take D = F and = −1), are said to be of type I, and pairs of type (b) are said to be of type II. Type I pairs are embedded into symplectic groups in roughly the same way as symplectic/orthogonal pairs are. Since we will not discuss type II pairs further, we will not say how they are embedded into symplectic groups. For any dual reductive pair (G; G0) in Sp(W ), we can match representations of G and representations of G0 using the Weil representation. More precisely, we say that an irreducible admissible G-representation π corresponds to an irreducible admissible G0-representation π0 if there is a nontrivial G × G0-map 0 0 ! ! π ⊗ π . This correspondence π 7! θ(π; V ) is called the local theta correspondence, and it was proven by Howe and Waldspurger to be 1-1 (see [13]). Note that in the case we care about (symplectic/even orthogonal), the local theta correspondence matches representations of Sp(V ) and representa- tions of O(V 0), due to the fact that the Weil representation pulls back to a representation of Sp(V ) × O(V 0).
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