Degrees and formal degrees for division algebras and GL_nover ap-adic field

Abstract

We compute in the tame case, the degrees of the irreducible representations of a division algebra and the formal degrees of the discrete series of GL(w) over a p-adic field and compare them.

A construction of the supercuspidal representations of ${\rm GL}\sb n(F),\;F\;p$-adic

“…We also compute the formal degrees of the representations n . These latter computations are similar to those in [6]. They serve the purpose of showing that supercuspidals constructed using ZeKe and those constructed using Ze~Ke~ , with c~ ^ e, are distinct.…”

“…Finally, we compute the formal degree of the supercuspidal n induced from t . The calculations are similar to those in §2 of [6]. We can now compute the formal degree of the supercuspidal it induced from t .…”

“…But 30nh° must commute with an~j, as we saw above; from (6) and (8), we can write w = eotf^^l +ein¡ + £o£(mt*(e,+i)Y…”

“…to apply the lemma), we can find an element go s W~_x such that m = g~xgo £ Kx. In fact, Lemmas 3.1 and 3.3 and Theorem 6.1 (6) show that for y G 77*, mym~x £ 770. Thus for any y G Ga_» n K¡~1 , g0_1y<?o £ 77* <-> y G 770.…”

Fields of rationality of cusp forms

“…], [Shi12], and [ST12] to the fixed-central-character setting. Properties (i) and (ii) will be necessary to apply the computations of [CMS90] and [AP05] to the fixed-central-character setting.…”