Buildings in the representation theory over local number fields
The paper starts off from the \(p\)-adic counterpart of semisimple real Lie groups modulo maximal compact subgroups, i.e, of symmetric spaces. As for an example, on the one hand one has \(SL_2(\mathbb{R})/SO(2)\simeq\mathbb{H}\), and on the other, \(SL_2(\mathbb{Q}_p)/SL_2(\mathbb{Z}_p)\), say, which, contrary to \(\mathbb{H}\), does not have an interesting topology yet. A second difference is that in the real case all maximal compact subgroups are conjugate, but no longer so in the \(p\)-adic situation. There arises the problem of classifying the conjugacy classes and of understanding the geometry involved. This is tackled by means of the Bruhat-Tits-building \(X\), which is a topological space carrying a cell structure, a metric and a compatible \(G\)-action, where now \(G\) is a connected reductive algebraic group over \(\mathbb{Q}_p\). The inner structure of \(G\) is reflected in its action on \(X\): (1) At each vertex \(x\) of \(X\) there exists a model of \(G\) over \(\mathbb{Z}_p\), i.e., a certain group scheme \(G_x\) over \(\mathbb{Z}_p\) such that e.g. the structure of \(X\) in a neighbourhood of \(x\) is given by the structure of the finite group \(G_x(\mathbb{F}_p)\). (2) The apartments of \(X\) are isometric to \(\mathbb{R}^d\) with \(d\) denoting the semisimple \(\mathbb{Q}_p\)-rank of \(G\). Illustrative examples and pictures are provided \((G=SL_2,SL_3,Sp_4)\). The main topic of the paper regards a description of the smooth representation theory of \(G\) by means of the building \(X\); this is a report (without proofs) on joint work with Stuhler. The interest in the representation theory of reductive \(p\)-adic groups comes from the Langlands programme, which asks for a parametrization of the finite-dimensional representations of the absolute Galois group \(G_{\mathbb{Q}_p}\) of \(\mathbb{Q}_p\) in terms of the smooth irreducible representations of \(Gl_n(\mathbb{Q}_p)\), \(n\geq 1\). As has been shown by \textit{C. J. Bushnell} and \textit{P. C. Kutzko} [Ann. Math. Stud. 129 (1993; Zbl 0787.22016)], the smooth representations of \(GL_n(\mathbb{Q}_p)\) can be characterized by the representations of the maximal compact subgroups. Here now it is shown that for general (semisimple) \(G\) the building \(X\) can be used to define, given a smooth representation \(V\) of finite length, a certain sheaf \(\widetilde V\) on \(X\) which, in turn, gives back \(V\). Finally, a trace formula in terms of the cohomology of \(\widetilde V\) is stated that allows for the computation of the values of the function \(\theta_V\) on the elliptic part of \(G\), which appears in the character function to \(V:\text{Tr}(\varphi,V)= \int_G\varphi(g)\theta_V(g)dg\) (for all elements \(\varphi\) of the Hecke algebra of \(G\)).
scientific article; zbMATH DE number 941019