×

zbMATH — the first resource for mathematics

Buildings in the representation theory over local number fields. (Gebäude in der Darstellungstheorie über lokalen Zahlkörpern.) (German) Zbl 0872.11028
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 C. J. Bushnell and 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 \(G\)).

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
PDF BibTeX XML Cite