Weil Representation, Howe Duality, and the Theta correspondence

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additional problems due to non-compactness, see[R-S].The 3-dimensional representation of O(V ) ∼ = D ∗ k/k ∗ × {±1} can be identifiedto the representation of D ∗ k/k ∗ × {±1} on the Schwartz space of functions on thevector space of trace 0 elements of D k on which D ∗ k acts by conjugation, and {±1}by multiplication.For a quadratic field K/k, let NK ∗ be the normaliser of K ∗ in D ∗ k, and let φ Kbe the character of order 2 of NK ∗ which is trivial on K ∗ . Let H K be the index-2subgroup of NK ∗ × {±1} given by the kernel of the homomorphism φ K × {±1} toC ∗ .The orbits for the action of D ∗ k/k ∗ × {±1} on the trace zero elements of D k are,besides the origin, of the form [D ∗ k/k ∗ × {±1}]/H K for a quadratic field extensionK/k. It follows from the Mackey theory that the only representations of O(V )appearing in S(V ) are those representations of D ∗ k/k ∗ × {±1} which are trivial onH K for some quadratic extension K/k. Therefore to prove that the map taking anirreducible representation of D ∗ k/k ∗ × {±1} appearing in S(V ) to its restriction toD ∗ k/k ∗ is an injection, it suffices to prove thatHom D ∗k[Ind D∗ kNK ∗1, IndD∗ kNL ∗φ L] = 0.We therefore have to prove that there are no non-zero functions f on D ∗ k such thatf(agb) = φ L (b)f(g) ∀a ∈ NK ∗ , g ∈ D ∗ k, b ∈ NL ∗ .To prove this, let j K (resp. j L ) be an element of NK ∗ (resp. NL ∗ ) which is notin K (resp L), and observe that for any g ∈ D ∗ k, j K · K ∩ g(j L · L)g −1 ≠ 0. This isbecause j K · K and g(j L · L)g −1 are 2-dimensional k-vector spaces, contained in the3-dimensional space of trace zero elements of D k . Therefore for all g ∈ D ∗ k, thereexists k ′ ∈ j K · K, l ′ ∈ j L · L such that gl ′ g −1 = k ′ , or g = k ′ gl ′−1 . Thereforef(g) = f(k ′ gl ′−1 ) = φ L (l ′−1 )f(g) = −f(g) = 0.Similarly to prove that the map taking an irreducible representation of D ∗ k/k ∗ ×{±1} appearing in S(V ) to its restriction to D ∗ k/k ∗ is a surjection, it suffices toprove that given any irreducible representation of D ∗ k, there exists a quadratic fieldextension K/k such that the representation has a K ∗ -invariant vector. This is awell-known fact about representations of quaternion division algebras.14
4.6.4 : dim V = 4. Suppose that V does not represent any zeros. In that caseSO(V ) = [D ∗ × D ∗ ] 1 = {(d 1 , d 2 ) ∈ D ∗ × D ∗ |Nd 1 = Nd 2 },where D is the unique quaternion division algebra over k, which can be identifiedto V as a quadratic space over k (with the quadratic form on V getting identifiedto the norm form on D), and the action of (d 1 , d 2 ) ∈ [D ∗ × D ∗ ] 1is by x →d 1 xd −12 . Let GSO(V ) = D ∗ × D ∗ , with the natural action on D. Clearly there areexactly 2 orbits for the action of GSO(V ) = D ∗ × D ∗ on D, one consisting of theorigin only, and the other all of D ∗ . Therefore from the Mackey theory, the onlyrepresentations of D ∗ × D ∗ appearing in its natural action on S(D) are of the typeW ⊗ W ∗ for W an irreducible representation of D ∗ . The representation W ⊗ W ∗ ofGSO(V ) extends in two ways to a representation of GO(V ), the group of orthogonalsimilitudes, and exactly one of these appears in S(D). The representations of O(V )appearing in S(D) are the restrictions of these representations. To conclude, forevery irreducible representation of D ∗ , one can associate a sum of representationsof O(V ) each of which occurs in the Howe correspondence with SL 2 , and converselyevery representation of O(V ) which occurs in the Howe correspondence with SL 2occurs in this way.4.6.5 : The Howe duality conjecture is specially simple to state (though notto prove!) for a dual reductive pair of type 2. Take for example the dual pair(GL(n, k), GL(m, k)). In this case one is looking at the representation of GL(n, k)×GL(m, k) on S(M(n, m; k)), where M(n, m; k) is the space of n × m matrices overk with the action of (A, B) ∈ GL(n, k) × GL(m, k) on X ∈ M(n, m; k) given byAX t B (this is the restriction of the metaplectic representation of Sp(nm, k) toGL(n, k) × GL(m, k) up to twisting by a character). The Howe duality conjecturein this case states that given an irreducible representation π of GL(n, k), thereexists a unique irreducible representation π ′ of GL(m, k) such that π ⊗ π ′ is aquotient of the GL(n, k) × GL(m, k) module S(M(n, m; k)). In this case a veryexplicit form of the Howe duality conjecture is expected, which as far as this authorcould find out, is not known. If n ≤ m, and if σ π and σ π ′are the representationsof the Weil-Deligne group associated by the Langlands correspondence to π and π ′respectively, then one expects that σ π ′= ν m−n2 σ ∗ π ⊕ ν − n 2 , where ν is the character|x| on k ∗ . In particular, if m = n, one expects that the Howe correspondence issimply taking the dual representation.15
Page 1 and 2: Weil Representation, Howe Duality,
Page 3 and 4: Let A ⊂ W be a closed subgroup an
Page 5 and 6: Moreover, the two sheeted covering
Page 7 and 8: In particular for K, the stabiliser
Page 9 and 10: 3 Dual Reductive PairsA dual reduct
Page 11 and 12: Now for a closed subgroup H ⊂ Sp(
Page 16 and 17: Remark 4.7: Howe correspondence for
Page 18 and 19: as (g, K) × (g ′ , K ′ )-modul
Page 20 and 21: etween the representations of their
Page 22 and 23: Definition 7.1: A pair (G, G ′ )
Page 24 and 25: Let W = W 1 ⊕ W 2 be a complete p
Page 26 and 27: f 2 belong to the same irreducible
Page 28 and 29: (b) The theta lift of an automorphi
Page 30 and 31: for all the local components π v o
Page 32: [Wa1] J.L.Waldspurger,“Correspond

Weil Representation, Howe Duality, and the Theta correspondence

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