Complement to the appendix of: “On the Howe duality conjecture”

Abstract

Abstract. Let F be a local field, nonarchimedean and of characteristic not 2. Let (V, Q) be a nondegenerate quadratic space over F, of dimension n. Let M r be the direct sum of r copies of V . We prove that, for r < n there is no nonzero distribution on M r which under the action of the orthogonal group transforms according to the character determinant.Let F be a local field, nonarchimedean and of characteristic not 2. Let V be a vector space over F of finite dimension n ≥ 1 equipped with a nondegenerate quadratic form Q. Let G = O(Q) be the orthogonal group of Q and χ the character of G such that χ(g) = 1(resp. χ(g) = −1) if the determinant of g is 1 (resp. −1).Let r ≥ 1 be an integer and let M r be the direct sum of r copies of V or equivalently the set of linear maps from F r onto V or, choosing a basis of V , the space of n × r matrices with coefficients in F. We let G act on M r on the left and GL r (F) act on the right. Finally, let S(M r ) be the space of locally constant functions on M r with compact support and values in C and its dual S (M r ) the space of distributions on M r . It will be convenient to include the case r = 0 with M 0 = (0) and the trivial action of G. Theorem. Suppose that r < n. If T ∈ S (M r ) transforms according to the character χ under the action of G = O(Q), then T = 0.This result was stated in Appendix 2 in [R] and used in the proof of the theorem in Chapter II. However, the proof given there was confusing to the reader due to incomplete detail. As the result itself is interesting we present here, in this short note, a full version of our proof.Note that this is false if r ≥ n. Indeed, suppose first that r = n and let x = (ξ 1 , . . . , ξ n ) be an orthogonal basis of V . Then Q(ξ i ) = α i = 0 (the quadratic form is nondegenerate) and by Witt Theorem the orbit of x under G is the set of all orthogonal systems (η i , . . . , η n ) such that for all i, Q(η i ) = α i . In particular, it is a closed orbit. Now the isotropy subgroup of x in G is trivial so that the orbit is homeomorphic to G. Choose a Haar measure dg on G. The distribution χ(g)dg is a nontrivial distribution on G transforming according to the character χ. We may view this distribution as a distribution on Gx, hence on V . If r > n we note that M n is imbedded into M r by (ξ 1 , . . . , ξ n ) → (ξ 1 , . . . , ξ n , 0, . . . , 0) as a closed invariant subset and, therefore, S (M n ) ⊆ S (M r ).

Theta functions, fourth moments of eigenforms and the sup-norm problem II

“…The fact that 𝜑 * Φ is square-integrable and cuspidal is trivial whenever G is anisotropic. If 𝐺 PGL 2 is split, then this follows from Rallis' tower property [Ral84,Moeg97] and the fact that the theta transfer of any cuspidal automorphic representation of SL 2 Sp 2 to the orthogonal group of the hyperbolic plane O(1, 1) vanishes.5 That the lift of 𝜑 ⊗ 𝜑 is cuspidal follows similarly, except that we need to use the theta transfer from O det to SL 2 , that is, we need first to lift 𝜄 (1) * 𝜑 ⊗ 𝜑 to O det . For that purpose, we use the homomorphism 𝜄 : 𝑀 → O det , which is the composition of the isomorphism 𝑀 SO det with the embedding SO det ↩→ O det .…”

“…The integral in Equation (A.6) is a theta lift of a cuspidal function in to . In this case, [Ral84] verifies that the theta lift of a cuspidal function to is cuspidal.…”

“…The fact that is square-integrable and cuspidal is trivial whenever G is anisotropic. If is split, then this follows from Rallis’ tower property [Ral84, Mœg97] and the fact that the theta transfer of any cuspidal automorphic representation of to the orthogonal group of the hyperbolic plane vanishes 5…”