Gopal Prasad

Strong rigidity of Q-rank 1 lattices

A discrete subgroup F of a locally compact topological group G is said to be a lattice in G if the homogeneous space G/F carries a finite G-invariant measure. A lattice F in G is said to be uniform (or co-compact) if G/F is compact; otherwise it is said to be non-uniform. A lattice F in a linear semi-simple group is said to be irreducible if no subgroup of F of finite index is a direct product of two infinite normal subgroups. Let G be a linear analytic semi-simple group which has trivial center and no compact factors, given a lattice A in G it is known that G decomposes into a direct p r o d u c t / / G i , such that for all i, Gi is a normal analytic subgroup of G; Ai=Ac~Gi is an irreducible lattice in G~ and llAi is a subgroup of A of finite index. Furthermore, if G = 1~ Gj is any decom-

Unrefined minimal K–types for p–adic groups

via their restriction to compact open subgroups was begun by Mautner, Shalika and Tanaka for groups of type AI. In contrast to real reductive groups where the representation theory of a maximal compact subgroup K is given in terms of slight modifications to Cartans theory of the highest weight, the representation theory of a (compact) parahoric subgroup ,~ is quite complicated. There is still no comprehensive theory for classifying the irreducible representa- tions. In the case of

Jacquet functors and unrefined minimal K-types

The notion of an unrefined minimal K-type is extended to an arbitrary reductive group over a non archimedean local field. This allows one to define the depth of a representation. The relationship between unrefined minimal K-types and the functors of Jacquet is determined. Analogues of fundamental results of Borel are proved for representations of depth zero.

Fake projective planes

A fake projective plane is a complex surface different from but has the same Betti numbers as the complex projective plane. It is a complex hyperbolic space form and has the smallest Euler Poincare characteristic among smooth surfaces of general type. The first example was constructed by Mumford. Later on two more examples were found by Ishida and Kato. A fourth possible one was recently proposed by Keum. In a recent joint work with Gopal Prasad, we showed that there are seventeen non-empty classes of fake projective planes and there can at most be four more specific classes. Higher dimensional analogues and examples were also obtained. The main purpose of the talk is to explain the joint work with Prasad and other related results such as arithmeticity of the lattices involved obtained earlier by Klingler and Yeung independently.

Volumes of S-arithmetic quotients of semi-simple groups

© Publications mathématiques de l’I.H.É.S., 1989, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Strong approximation for semi-simple groups over function fields

Let K be a field and A be a K-algebra. For a variety V, defined over K, we shall let V(A) denote the set of A-rational points of V. In case A is a locally compact topological ring, V(A) has a natural locally compact Hausdorff topology induced by the topology on A (see Weil [25: App. III]); in the sequel we shall assume V(A) endowed with this topology. If K is a local field (i.e., a non-discrete locally compact field) and V is a smooth K-variety, then V(K) is a K-analytic manifold. Moreover, when V is a K-group, V(K) is a K-analytic group. Let G be a connected semi-simple affine algebraic group defined, isotropic and almost simple over a local field K of arbitrary characteristic. Let G = G(K). Let G+ be the normal subgroup of G generated by the K-rational points of the unipotent radicals of parabolic K-subgroups of G. The object of this paper is to prove the following:

Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric spaces.

Computation of the metaplectic kernel

© Publications mathématiques de l’I.H.É.S., 1996, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the congruence subgroup problem: determination of the "Metaplectic Kernel"

Let F be a global field (i.e. either a number field or a function field in one variable over a finite field) and let oo be the set of its archimedean places. Let G be a finite non-empty set of places of F containing oo. Let o = o ( ~ ) denote the ring of G-integers of F. Let A(~) denote the ring of G-adeles i.e. the restricted direct product of the completions F v for r eG. Let f# be an absolutely simple*, simply connected subgroup of SL, defined over F. Recall that a subgroup F of C#(F) is an G-arithmetic subgroup if FnSL(n, o) has finite index in F as well as in ~(o):=fY(F)c~SL(n,o). An G-arithmetic subgroup F is a ~congruence subgroup if there exists a non-zero ideal a in o (= o(G)) such that

ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS

In a recent paper we have classified fake projective planes. Natural higher dimensional generalization of these surfaces are arithmetic fake