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QUATERNION DISTINGUISHED GENERIC REPRESENTATIONS OF GL2n BY

Miyu Suzuki Faculty of Mathematics and Physics, Institute of Science and Engineering Kanazawa University, Kakumamachi, Ishikawa, 920-1192, Japan e-mail: miyu-suzuki@staff.kanazawa-u.ac.jp

ABSTRACT

Let E/F be a quadratic extension of non-Archimedean local fields of characteristic 0. Let D be the unique quaternion division algebra over F and fix an embedding of E to D. Then, GLm (D) can be regarded as a subgroup of GL2m (E). Using the method of Matringe, we classify irreducible generic GLm (D)-distinguished representations of GL2m (E) in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil–Deligne group of E, we prove a sufficient condition for a generic representation in the image of the unstable base change lift from the unitary group U2m to be GLm (D)-distinguished.

Introduction Let F be a non-Archimedean local field of characteristic 0. Let G, H be algebraic groups over F and suppose H is a closed subgroup of G. A smooth representation π of G(F ) is said to be H(F )-distinguished if it has a nonzero H(F )-invariant linear form. This paper concentrates on the case of (G, H) = (ResE/F GL2m , GLm (D)), where E is a quadratic extension of F , ResE/F denotes the restriction of scalars and D is the unique quaternion division algebra over F . Given an embedding of E to D, H can be regarded as a subgroup of G. We classify irreducible smooth generic H(F )-distinguished representations of G(F ) and relate them to a functorial lift from the quasi-split unitary group U2m . Received December 19, 2018 and in revised form August 14, 2019

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M. SUZUKI

Isr. J. Math.

In order to explain the background, we consider the global setting. Let E/F be a quadratic extension of number fields and AE , AF the rings of adeles of E and F , respectively. Let π be an irreducible cuspidal automorphic representation of GL2m (AE ). Then, π is said to be GL2m -distinguished if there is a cusp form f in the space of π which has a nonzero period integral over GL2m :  f (h) dh = 0. Z(AF )GL2m (F )\GL2m (AF )

Here, Z denotes the center of GL2m . Flicker and Rallis (see [F]) conjectured that π is GL2m -distinguished if and only if it is an unstable base change lift of a generic cuspidal automorphic representation of the quasi-split unitary group U2m (AF ). Let D be a quaternion algebra over F which E embeds. Then, distinguished cuspidal representations of GL2m (AE ) with respect to GLm (D) are defined similarly as above. In [S], the author defined a non-degenerate character θτ associated to D for the quasi-split unitary group and conjectured that π is GLm (D)distinguished if and only if it is an unstable base change lift of a θ1 -generic and θτ -generic cuspidal representation of U2m (AF ). This conjecture slightly generalizes that of Flicker and Rallis. Now, we go back to the local setting. Recall that G = ResE/F GL2m . Let W DF be the Weil–Deligne group of F and L G, L U2m be the L-groups of G and U2m . For an irreducible sm