# Journal of the Ramanujan Mathematical Society

Volume 31, Issue 2, June 2016 pp. 109–124.

Base change and (GL n (F), GL n-1 (F))-distinction

**Authors**:
Arnab Mitra and C. G. Venketasubramanian

**Author institution:**Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

**Summary: **
Let F be a nonarchimedean local field and E a finite cyclic
extension of F of prime degree d. Further let G n-1 be
embedded into G n as block matrices in the usual way. It is not
true in general that the base change lift of a
G n-1 (F)-distinguished representation of G n(F) is
G n-1 (E)-distinguished. We obtain a precise condition for an
irreducible G n-1 (F)-distinguished representation
π of G n(F) to be taken to a G n-1 (E)-distinguished
representation by the base change map. If π is unitarizable
and G n-1 (F)-distinguished, then we show that the base change
lift of π is G n-1 (E)-distinguished. We then analyse the
fiber of the base change map over a G n-1 (E)-distinguished
representation of G n(E) and determine the number of
G n-1 (F)-distinguished representations in the
fiber.

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