Distinction of the Steinberg representation for inner forms of GL(n)

2016/02/16 by Matringe, Nadir
#22E50 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1602.05101

Abstract

Let F be a non archimedean local field of characteristic not 2. Let D be a division algebra of dimension d² over its center F, and E a quadratic extension of F. If m is a positive integer, to a character χ of E^*, one can attach the Steinberg representation St(χ) of G=GL(m,D⊗_F E). Let H be the group GL(m,D), we prove that St(χ) is H-distinguished if and only if χ_|F^* is the quadratic character η_E/F^md-1, where η_E/F is the character of F^* with kernel the norms of E^*. We also get multiplicity one for the space of invariant linear forms. As a corollary, we see that the Jacquet-Langlands correspondence preserves distinction for Steinberg representations.

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