Epsilon factors as algebraic characters on the smooth dual of GLn\mathrm{GL}_n

Let $K$ be a non-archimedean local field and let $G = \mathrm{GL}_n(K)$. We have shown in previous work that the smooth dual $\mathbf{Irr}(G)$ admits a complex structure: in this article we show how the epsilon factors interface with this complex structure. The epsilon factors, up to a constant term, factor as invariant characters through the corresponding complex tori. For the arithmetically unramified smooth dual of $\mathrm{GL}_n$, we provide explicit formulas for the invariant characters.Comment: 12 pages. Minor improvements, new titl

Reduced C*-algebra of the p-adic group GL(n) II

Let G be the p-adic group GL(n). Using C*-algebra techniques, we obtain a very explicit description of the tempered dual of G in terms of Bernstein parameters and extended quotients. We also prove that Plancherel measure (on the tempered dual of a reductive p-adic group) is rotation-invariant.Comment: 15 page

Base change and K-theory for GL(n,R)

We investigate base change $C/R$ at the level of $K$-theory for the general linear group $GL(n,R)$. In the course of this study, we compute in detail the $C*$-algebra $K$-theory of this disconnected group. We investigate the interaction of base change with the Baum-Connes correspondence for $GL(n,R)$ and $GL(n,C)$. This article is the archimedean companion of our previous article in the Journal of Noncommutative Geometry.Comment: 17 pages, introduction and section 5 completely rewritte

L-packets and depth for SL_2(K) with K a local function field of characteristic 2

Let G = SL_2(K) with K a local function field of characteristic 2. We review Artin-Schreier theory for the field K, and show that this leads to a parametrization of certain L-packets in the smooth dual of G. We relate this to a recent geometric conjecture. The L-packets in the principal series are parametrized by quadratic extensions, and the supercuspidal L-packets of cardinality 4 are parametrized by biquadratic extensions. Each supercuspidal packet of cardinality 4 is accompanied by a singleton packet for SL_1(D). We compute the depths of the irreducible constituents of all these L-packets for SL_2(K) and its inner form SL_1(D).Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1302.603

K-theory and the connection index

Let G denote a split simply connected almost simple p-adic group. The classical example is the special linear group SL(n). We study the K-theory of the unramified unitary principal series of G and prove that the rank of K_0 is the connection index f(G). We relate this result to a recent refinement of the Baum-Connes conjecture, and show explicitly how generators of K_0 contribute to the K-theory of the Iwahori C*-algebra I(G).Comment: 11 page

Base change and K-theory for GL(n)

Let F be a nonarchimedean local field and let G = GL(n) = GL(n,F). Let E/F be a finite Galois extension. We investigate base change E/F at two levels: at the level of algebraic varieties, and at the level of K-theory. We put special emphasis on the representations with Iwahori fixed vectors, and the tempered spectrum of GL(1) and GL(2). In this context, the prominent arithmetic invariant is the residue degree f(E/F).Comment: 20 pages. Completely rewritten, much more concis