LECTURE 16: UNITARY REPRESENTATIONS LECTURE BY SHEELA DEVADAS STANFORD NUMBER THEORY LEARNING SEMINAR FEBRUARY 14, 2018 NOTES BY DAN DORE

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1 LECTURE 16: UNITARY REPRESENTATIONS LECTURE BY SHEELA DEVADAS STANFORD NUMBER THEORY LEARNING SEMINAR FEBRUARY 14, 2018 NOTES BY DAN DORE Let F be a local field with valuation ring O F, and G F the group GL 2 pf q. If pπ, V q is an admissible representation of G F, we define: Definition 1. pπ, V q is pre-unitary if it admits a positive definite G F -invariant Hermitian form. We say pre-unitary rather than unitary, since V may not be complete. However, given a preunitary representation pπ, V q, we may complete V to a Hilbert space p V and extend the action of π to pπ. This preserves the notion of irreducibility in an appropriate sense: Lemma 2. If pπ, V q is a pre-unitary admissible representation, then pπ, V q is algebraically irreducible if and only if ppπ, p V q is topologically irreducible: i.e. it has no non-trivial invariant closed subspaces. Proof. If we let ρ be a (finite dimensional) irreducible representation of GL 2 po F q, then we write V pρq for the set of all vectors v P V which transform under GL 2 po F q according to ρ, i.e. the span of all of the copies of ρ inside Res G F GL 2 po F q V. We may define V p pρq similarly. We will show that V pρq V p pρq. By admissibility of V, V À ρ V pρq, and V pρq are mutually orthogonal and finite-dimensional. This implies that: pv xà ρ V pρq, the Hilbert direct sum of the V pρq. Thus, p V pρq V pρq. Hence V is the set of all GL 2 po F q-finite vectors in p V. Because GL 2 po F q is compact, GL 2 po F q- finite vectors are dense in every closed invariant subspace of p V. Therefore V X W 1 is dense in any G F -invariant closed subspace W 1 of p V. Thus, any closed invariant subspace W 1 Ď p V is the closure of V X W 1. If V is algebraically irreducible, this means that V X W 1 0 or V X W 1 V, so W 1 0 or W 1 p V. Therefore if pπ, V q is algebraically irreducible, then ppπ, p V q is topologically irreducible. On the other hand, we need to show that if V 0 Ď V is a nontrivial proper invariant subspace, then V 0 Ĺ p V. For V 0 an invariant subspace of V, we see that if we let V 0 pρq V pρq X V 0, then V 0 À ρ V 0pρq. Since we also have p V xà ρ p V pρq (recall p V pρq V pρq), we see that the closure V 0 of V 0 in p V must be V 0 xà ρ V 0pρq. Thus V 0 is invariant and is nontrivial exactly when V 0 is, so we see that if p V is topologically irreducible, V 0 0 or V 0 p V, so V 0 0 or V 0 V. Therefore pπ, V q is algebraically irreducible if ppπ, p V q is topologically irreducible. Remark 3. It is not a priori obvious that any topologically irreducible Hilbert space representation pv of G F arises in this way from an irreducible admissible representation of G F. In other words, why should the space of smooth vectors (i.e. the vectors which are fixed by an open subgroup of G F ) of p V be non-zero? 1

2 Theorem 4. Let pπ, V q be an infinite-dimensional 1 irreducible admissible representation of G F. It is pre-unitary exactly in these cases: (1) π is super-cuspidal and the central character ω π satisfies ω π ptq 1 for all t. (2) π π µ1,µ 2 is a non-special principal series with µ 1, µ 2 unitary characters. (3) π π µ1,µ 2 is a non-special principal series with µ 2 µ 1 1 and µ : µ 1 µ 1 2 x σ for 0 ă σ ă 1. (4) π π µ1,µ 2 is a special principal series such that µ 1 pxq x 1{2 χpxq, µ 2 pxq x 1{2 χpxq for χ a unitary character. Furthermore, on the Kirillov model Kpπq of pπ, V q, the invariant scalar product takes the form pξ, ηq ξpxqηpxqdx. The rest of these notes concern the proof of this theorem. 1 Super-cuspidal case First, we will consider case (1), where π is super-cuspidal. The condition that ω π ptq 1 is necessary because for any ξ, η P V, we have by G F -invariance of the Hermitian pairing p, q on V : pξ, ηq π`p 0 t 0 t q ξ, π`p 0 t 0 t q η ω π ptq ω π ptq pξ, ηq. Thus, by non-degeneracy, ω π ptq 1 for all t. Now, assume that this condition holds. Fix ζ 0 P ˇV a non-zero vector. For any ξ, η P V, we consider the following function on G F : f ξ,η : g ÞÑ xπpgqξ, ζ 0 yxπpgqη, ζ 0 y. Here, x, y is the canonical pairing on V ˇV. Then f ξ,η is a function from G F to C which is compactly supported modulo the center Z F of G F - this follows from the definition of supercuspidality, because f ξ,η is a product of matrix coefficients. (See Theorem 3 in [1].) Now, it makes sense to integrate over G F to obtain a Hermitian pairing: pξ, ηq : xπpgqξ, ζ 0 yxπpgqη, ζ 0 y dg. G F {Z F Note that the integrand is a Z F -invariant function because the central character is unitary by assumption. It is not hard to see that this is G F -invariant by unimodularity of G F. Now, we must check that this is positive-definite. Assume that pξ, ξq 0. This means that for all g, πpgqξ is orthogonal to ζ 0, or equivalently (by G F -invariance of x, y) that ξ is orthogonal to ˇπpgqζ 0. Since ˇπ is irreducible, this implies that ˇπpGq ζ 0 generates ˇV and therefore that ξ 0. 1 so it does not factor through the determinant 2

3 Next, we will describe this pairing on the Kirillov model. We claim that it is given by: pξ, ηq ξpxqηpxq dx. It suffices to show that this gives an invariant inner product on Kpπq, since these are unique up to a scalar: an invariant inner product on pπ, V q is determined by the associated isomorphism π ÝÑ ˇπ, so two distinct invariant inner products differ by an automorphism of the irreducible G F -module ˇV. This must be a scalar by Schur s lemma. The pairing defined above is clearly a positive-definite Hermitian inner product, so it suffices to show that the pairing is G F -invariant. It actually suffices to check this invariance under the family of operators F a,b : rx ÞÑ ξpxqs ÞÑ rx ÞÑ ψ F pbxqξpaxqs: ψ F pbxqξpaxqψ F pbxqηpaxqdx ψ F pbxqξpxqψ F pbxqηpxqdx ξpxqηpxqdx. since we assume ψ F unitary. This gives invariance under the mirabolic subgroup H t` a b u by the definition of the Kirillov model. Since we are in the supercuspidal case, Kpπq S pf q. In fact the mirabolic subgroup acts irreducibly on S pf q. Since we know that a G F -invariant (hence H-invariant) bilinear form exists by the previous discussion, we see that up to a nonzero constant, this G F -invariant bilinear form must equal pξ, ηq ξpxqηpxq dx as we desired to show. This completes the case of super-cuspidal representations, so it suffices to consider the case of principal series. 2 Principal series case Let s assume that pπ, V q is a pre-unitary irreducible admissible representation. We have a complex semi-linear map J from V to ˇV defined by : xξ, Jηy pξ, ηq. This gives an isomorphism from π to ˇπ. Now, assume that π π µ1,µ 2 is a non-special principal series representation. This is defined on the space B µ1,µ 2 of locally constant function ϕ on G F such that: ϕ`p a 0 b q g µ 1 paqµ 2 pbq a{b 1{2 ϕpgq. Now, we have an isomorphism B µ1,µ 2 ÝÑ B µ1,µ 2» π sending ϕ to ϕ. We know from a previous lecture that ˇπ» B µ 1,µ Now, there are two cases, corresponding to cases (2) and (3) in the statement of the theorem. The first of these is the case that µ 1 1 µ 1, µ 1 2 µ 2, and this means that µ 1, µ 2 are unitary. The second is the case that µ 1 1 µ 2, µ 1 2 µ 1. 3

4 Since B µ1,µ 2» B λ1,λ 2 iff tµ 1, µ 2 u tλ 1, λ 2 u as unordered pairs, these possibilities are necessary and sufficient for π and ˇπ to be G F -isomorphic. As we saw above, this is necessary for π to be pre-unitary, but it is not clearly sufficient: an isomorphism from π to ˇπ induces a non-degenerate Hermitian invariant bilinear pairing on V, but it is not a priori clear that it should be positive definite. 2.1 Case (2) In the first case, we define the bilinear pairing by: pϕ 1, ϕ 2 q : ϕ 1 pgqϕ 2 pgq dg B F zg F GL 2 po F q ϕ 1 pmqϕ 2 pmq dm. The second equality follows from the Cartan decomposition B F GL 2 po F q G F. The notes from Lecture 12 of this seminar ([5]) show that this integral defines a non-degenerate G F -invariant bilinear pairing (this was how we identified the contragredient of π µ1,µ 2 with π µ 1,µ 1). 1 2 To show that this is Hermitian and positive-definite, we will compute in the Kirillov model Kpπq. To ϕ P B µ1,µ 2, we associate ξ ϕ P Kpπq, defined by: ξ ϕ pxq : µ 2 pxq x 1{2 ϕ w 1 ` 1 y ψ F pxyq dy. Here, w is the Weyl group generator w ` 1 0. Now, we use the following identity, which comes from explicitly realizing Bruhat decomposition for GL 2 : ` a b c d c 1 det g w 1 0 c 1 d{c which is valid whenever c 0. This implies that: ϕ 1 pgqϕ 2 pgq dg ϕ 1`w 1 p 1 0 x 1 q ϕ 2`w 1 p 1 0 x 1 q dx. B F zg F Defining Φ i pxq : ϕ i`w 1 p 1 0 x 1 q, the above identity may be rephrased as: pϕ 1, ϕ 2 q Φ 1 pxqφ 2 pxq dx., We may use Fourier inversion to see: pϕ 1, ϕ 2 q Φ 1 pxq {Φ 2 pyqψ F pxyq dy dx xφ 1 pyq x Φ 2 pyq dy. We are able to do this because, as discussed in Section 1.9 of [1], Φ corresponding to ϕ P B µ1,µ 2 must be proportional to µpxq 1 x 1 for x large. Since µpxq : µ pxqµ pxq satisfies µpxq 1, we see that Φ is square integrable on F, so Φ p is its Fourier transform in the L 2 sense. Furthermore, we can determine the image of the map Φ Ñ Φ, p from which we are able to deduce that the Φ p are integrable and ΦpyqψF p pxyq dy Φpxq. 4

5 Therefore pϕ 1, ϕ 2 q xφ 1 pyq x Φ 2 pyq dy µ 2 pyq 1 y 1{2 ξ ϕ1 pyqµ 2 pyq 1 y 1{2 ξ ϕ2 pyqy ξ ϕ1 pxqξ ϕ2 pxqdx, using the relation dx x 1 dx and the fact that µ 2 is unitary. In this form, it is easy to check that the pairing is positive definite and Hermitian. This settles case (2), where µ 1, µ 2 are unitary. 2.2 Case (3) Now, we must settle case (3): π π µ1,µ 2 with µ 1 µ 1 2 and µ i µ 1 i (i.e. µ i not unitary). Since we know what characters of F must look like, this means that µ µ µ x σ for σ 0 (if σ 0, the µ i would be unitary). Without loss of generality, we may assume that σ ą 0 since we may switch µ 1 and µ 2 if desired. We will define an operator A: B µ1,µ 2 Ñ B µ1 1,µ 2 1 B µ2,µ 1 by: paϕqpgq : ϕ`w p 1 0 x 1 q g dx. For fixed g, the integrand ϕ`w p 1 x q g grows as µ 1 pxq x 1 x σ 1 for large x, so this integral converges (see p.1.28, [1]). We will show that A 0 and that A is G F -equivariant. To see that A is nonzero, we use the function fpgq fp` a b c d q : det g 1{2 c 1 µ 1 pcqµ 1 pdet gq for c 0 and fpgq 0 otherwise. It can be checked that f P B µ1,µ 2, and furthermore, pafqp1q 1. Hence A 0. To see that A is G F -equivariant, we note that for h, g P G F pπ µ2,µ 1 pgqaϕqphq paϕqphgq ϕ`w p 1 0 x 1 q hg dx pπ µ1,µ 2 ϕq`w p 1 0 x 1 q h dx so A is G F -equivariant as we desired to show. paπ µ1,µ 2 ϕqphq, 5

6 It follows that an invariant scalar product on B µ1,µ 2 must look like: pϕ 1, ϕ 2 q cxaϕ 1, ϕ 2 y for some constant c, since any invariant Hermitian product on B µ1,µ 2 will give an isomorphism between π µ1,µ 2 π µ1,µ 2 and its dual B µ2,µ 1, and those isomorphisms are unique up to a scalar by irreducibility of B µ2,µ 1. We will show that when σ ă 1, this actually defines a positive-definite Hermitian pairing. We have Aϕ 1 P B µ2,µ 1, and ϕ 2 P B µ1,µ 2 B µ 1,µ 1. For ϕ P B µ 2 1 1,µ 2, we define: Φpxq ϕ`w 1 p 1 x q, Φ 1 pxq paϕq`w 1 p 1 x q. Now, again using the Bruhat decomposition identity ` a b c d w 1 we may see that for g ` a b c d with c 0: c 1 det g 0 c 1 d{c, ϕpgq µ 1 pdet gq det g 1{2 µ 1 pcq c 1 Φpd{cq, since ϕ P B µ1,µ 2 transforms under upper triangular matrices in a specified way. Furthermore: Φ 1 pxq ϕpw ` 1 y w 1 p 1 0 x 1 qqdy 1 x ϕp` y 1`xy qdy µ 1 pyq y 1 Φp 1 ` xy qdy (by the Bruhat decomposition identity) y µ 1 pyqφpx ` y 1 qdy µpyqφpx ` yqdy, where in the last equality we make a change of variables y ÞÑ y 1. Now, if we define: pϕ 1, ϕ 2 q cxaϕ 1, ϕ 2 y, we may use the above identities to see that: pϕ 1, ϕ 2 q c paϕ 1 qpgqϕ 2 pgq dg B F zg F c paϕ q`w 1 1 p 1 0 x 1 q ϕ 2`w 1 p 1 0 x 1 q dx c Φ 1 1pxqΦ 2 pxq dx c Φ 1 px ` yqφ 2 pxq y σ dy dx. 6

7 For this to be positive-definite, we need to find some cpσq ą 0 such that: cpσq Φpx ` yqφpxq y σ dx dy ě 0 for any Φ defined as: Φpxq ϕ`w 1 p 1 x q for ϕ P B µ1,µ 2. The space of such Φ is the space F pµq of locally constant functions on F which are proportional to µpxq 1 x 1 for x " 0 (see Section 1.9 in [1]). Since the space of Schwarz functions S pfq are 0 for x " 0, we have S pf q Ď F pµq. Note that cpσq y σ dy cpσqy σ 1 dy. We note that gpyq cpσqy σ 1 must be a positive-definite function on F, meaning that for any n-tuple of elements ty i u n i1 in F, the matrix pgpy i y j qq must be positive Hermitian. This is because, as stated in Proposition 4.1, Chapter 1, [3], a function f 0 on a locally compact abelian group H is positive definite, if for any f P C c phq, we have f 0 pyq fpy xqfp xqdxdy ě 0, where dy is the Haar measure on that group. We see that this condition is exactly equivalent to our previous condition cpσq Φpx ` yqφpxq y σ dx dy ě 0. We will see that it is possible to choose such a cpσq iff σ ă 1. Then since cpσqy σ 1 is a positive definite function, the distributional Fourier transform of cpσq y σ dy cpσqy σ 1 dy must be a positive measure on F, by Bochner s theorem. (For proof of this theorem see p.19, [2].) This Fourier transform is proportional to x 1 σ dx. Since x 1 σ is only a locally L 1 function near 0 when σ ă 1, this condition is necessary for this to be a measure on F. Now, for 0 ă σ ă 1, the definition of the γ-factor implies that there is a constant γpσq for any Φ P S pf q such that: pφpyq y σ dy 1 Φpxq x 1 σ dx. γpσq This is because if we define L Φ pχ, sq F Φpxqχpxq x s dx, there is a factor γpχ, sq depending only on χ and s such that L Φ p χ, 1 sq γpχ, sql pφ pχ, sq. (See p.1.41 in [1], or discussion in previous lectures.) Thus, if we choose cpσq γpσq and define pϕ 1, ϕ 2 q cpσqxaϕ 1, ϕ 2 y, then we have: pϕ 1, ϕ 2 q γpσq Φ 1 px ` yqφ 2 pxq y σ dx dy. It remains to show that the inner product takes the desired form on the Kirillov model. Since the Fourier transform of y ÞÑ Φ 1 px ` yq is z ÞÑ Φ x 1 pzqψ F pxzq, we see that γpσq Φ 1 px ` yq y σ dy xφ 1 pzqψ F pxzq z 1 σ dz. We are able to perform the Fourier transform since we can describe the behavior of the Φ i for x large - Φ i pxq must be proportional to µpxq 1 x 1 x σ 1 (see p.1.31, [1]), and σ ą 0. 7

8 Then recalling the definition of ξ ϕ the corresponding element to ϕ in the Kirillov model, we see that pϕ 1, ϕ 2 q γpσq Φ 1 px ` yqφ 2 pxq y σ dx dy pφ 1 pzqψ F pxzqφ 2 pxq z 1 σ dx dz pφ 1 pzqψ F pxzqφ p 2 pzq z 1 σ dz ξ ϕ1 pzqξ ϕ2 pzq µ pzq 2 z 1 2 z 1 σ dz ξ ϕ1 pzqξ ϕ2 pzqdz, as we desired to show. We are able to perform the Fourier transforms since we can describe the behavior of the Φ i for x large (see p.1.31, [1]). 3 Special case Finally, we must settle the final case (4), where π π µ1,µ 2 is special. Without loss of generality, we may assume that µ µ µ x. If π is pre-unitary, π» ˇπ. As above, this implies that µ 1 µ 2 1 (we cannot have µ 1, µ 2 both unitary, since this is incompatible with the assumption that µ x ). One may show that in this case we have µ 1 pxq x 1{2 χpxq and µ 2 pxq x 1{2 χpxq, for χ a unitary character. The space of π is Bµ,µ 2 Ď B µ1,µ 2 defined by the condition that: ϕ`w 1 p 1 0 x 1 q dx 0. By invariance of B 0 µ 1,µ 2 as a subspace, if ϕ P B 0 µ 1,µ 2, then for any g P G F, we have: ϕ`w 1 p 1 x q g dx 0. Thus, we cannot use the formula for A that we used in the previous cases. Instead, we use a limit: pϕ 1, ϕ 2 q : lim γpσq Φ 1 px ` yqφ 2 pxq y σ dx dy lim xφ 1 pzqφ x 2 pzq z 1 σ dz. σñ1 σñ1 We must verify that the properties of Φ 1, Φ 2 make the Fourier inversion make sense, and make sure that the limit exists and has the desired properties. We can do this Fourier inversion because σ 1 implies that for x large, Φ 1, Φ 2 are proportional to µpxq 1 x 1 x 2, so they are in L 1 X L 2 and we can apply Parseval s formula. If ϕ P B 0 µ 1,µ 2, then p Φp0q 0. Because p Φ is in fact locally constant, and vanish outside some compact subset of F, that gives the desired convergence of the limit (see p.1.29, [1]). 8

9 Since the limit exists, we can check that: pϕ 1, ϕ 2 q xφ 1 pxqφ x 2 pxq dx ξ ϕ1 pxqξ ϕ2 pxq dx. This shows p, q is Hermitian and positive definite. Thus, we need to check invariance under B F and under w. The former comes from the formula for p, q on the Kirillov model. What remains is to check invariance under w. We have (see p.1.52, [1]) that γpσq 1 q σ 1 q σ 1. For ϕ P B 0 µ 1,µ 2, we have, via a change of variables: lim γpσq σñ1 Φpxq x y σ 1 1 q σ dx lim σñ1 1 q σ 1 Φpx ` yq x σ dx. We note that Φpx ` yq x dx Φpx ` yqdx 0 for all ϕ P B 0 µ 1,µ 2. Therefore lim σñ1 1 q σ 1 q σ 1 Φpx ` yq x σ p1 q σ q Φpx ` yqp x σ x q dx dx lim. σñ1 1 q σ 1 d Now if we observe that dσ x σ vpxq x σ log q (where vpxq is the valuation) and noting that the integral Φpx ` yqvpxqdx is absolutely convergent (to justify the derivation under the integral), we are able to apply L Hospital s rule to see that lim σñ1 Therefore Φpx ` yqp x σ x q dx 1 q σ 1 lim σñ1 lim γpσq σñ1 Φpxq x y σ 1 dx p1 q 1 q Φpx ` yqp vpxqq x σ log q dx. q σ 1 log q Φpx ` yqvpxqdx. Now, this allows us to show that: pϕ 1, ϕ 2 q lim γpσq Φ 1 px ` yqφ 2 pxq y σ dx dy σñ1 lim γpσq Φ 2 pxq Φ 1 px ` yq y σ dy dx σñ1 lim 1 1 Φ 2 pxq Φ 1 px ` yqvpyq dy dx σñ1 q 1 1 Φ 1 pxqφ 2 pyqvpx yq dx dy. q 9

10 Now, we may check invariance under the Weyl group element w: pπpwqϕ 1, πpwqϕ 2 q q ϕ 1`w 1 p 1 x q w ϕ 2 ϕ ` x 1 ϕ 2 ` 1 0 y 1 w 1 ` 1 y wvpx yq dx dy vpx yq dx dy Φ 1 p x 1 q x 2 Φ 2` y 1 y 2 vpx yq dx dy Φ 1 pxqφ 2 pyqvpx 1 y 1 q dx dy Φ 1 pxqφ 2 pyq `vpx yq vpxyq dx dy pϕ 1, ϕ 2 q q Φ 1 pxqφ 2 pyqvpxyq dx dy. Now, we want the second term to vanish. We have: Φ 1 pxqφ 2 pyqvpxyq dx dy Φ 2 pyqvpyq Φ 1 pxq dx dy ` Φ 1 pxqvpxq Φ 2 pyq dy dx. Both terms vanish by the defining condition that ϕ i P B 0 µ 1,µ 2 : this says exactly that Φ i pxq dx 0. Therefore we have invariance under the Weyl element w and hence under G F, which is what we desired to show. This completes the proof of the theorem for the case of special representations References [1] Godement, R. Notes on Jacquet-Langlands Theory, The Institute for Advanced Study, 1970, available at conrad/conversesem/refs/godement-ias.pdf. [2] Rudin, W. Fourier analysis on groups. New York : Interscience [3] Van Den Berg, C., and Gunnar Forst. Potential theory on locally compact abelian groups. Vol. 87. Springer Science & Business Media, [4] conrad/conversesem/notes/l11.pdf [5] conrad/conversesem/notes/l12.pdf 10