My initial work was concerned with the rigidity of lattices in semi -simple Lie groups over real and nonarchimedean local fields. I recall that a lattice in a locally compact group G is a discrete subgroup Gamma such that the homogeneous space G/Gamma carries a finite G-invariant Borel measure. A lattice is said to be rigid if it "determines" the ambient topological group. In my first paper [1], written jointly with M. S. Raghunathan, it was shown that the R-rank of a semi-simple Lie group can be determined from the abstract group structure of any lattice in it. Later I proved that non-cocompact lattices in semi-simple Lie groups, with a factor of R-rank 1, are strongly rigid ([2]). This, combined with the beautiful results of G. D. Mostow and G. A. Margulis on strong rigidity implies that any irreducible lattice in a real semi-simple Lie group G is strongly rigid if G is not locally isomorphic to a product of SL2 (R) and a compact group. Mostow had earlier shown that to prove strong rigidity it is enough to construct an equivariant pseudo-isometry between the associated symmetric spaces. The proof of existence of such a pseudo-isometry is fairly simple in case the lattice is cocompact, i.e. G/Gamma is compact. However, in case G/Gamma is non-compact, the existence of an equivariant pseudo-isometry is not completely obvious. In case G has a factor of R-rank 1, using the fundamental domains constructed by Borel-Harish Chandra (for arithmetic lattices) and by Garland -Raghunathan (for arbitrary lattices), I proved that the required pseudo-isometry exists.
After the above work was done, I began to investigate lattices in semi-simple groups over nonarchimedean local fields and also considered arithmetical questions on semi-simple groups defined over global fields. I proved that cocompact lattices in semi-simple algebraic groups defined over local fields are strongly rigid ([5]), but in case the local field is of positive characteristic, lattices are not necessarily infinitesimally rigid ([6]). The proof of strong rigidity makes use of the geometry of Bruhat-Tits buildings and Mostow's ideas in his proof of strong rigidity of lattices in real semi-simple Lie groups.
In 1976, I proved the strong approximation property for general semi-simple, simply connected, groups over global fields of arbitrary characteristics ([7]). For number fields, this was proved by M. Kneser and V. P. Platonov; however, their methods did not work if the global field is the function field of a curve over a finite field and my proof required ergodic theoretic considerations, as well as the Kneser-Tits property. The latter (i.e. the Knesar-Tits property) was proved by Platonov, and Raghunathan and I have given a very simple and conceptually a more satisfactory proof ([8]) which is likely to be of use in the study of semi-simple groups over general fields.
A good part of my research during the late seventies and early eighties was done in collaboration with M. S. Raghunathan and was devoted to determining the topological central extensions of semi-simple groups over local fields and the computation of the metaplectic kernel of any semi-simple, simply connected, isotropic group defined over a global field. Our approach to settle these problems is different from the earlier work in the area. It uses the structure and the geometry of the groups in a very intimate way and avoids generators and relations completely. Details of this work are given in papers [9, 10, 11], and a survey of the known results and open problems in the area can be found in my address at the International Congress of Mathematicians held at Kyoto in 1990. In collaboration with A. S. Rapinchuk, I have now computed the metaplectic kernel of all semi-simple, simply connected groups defined over a global field ([17]). Note that the computation of the metaplectic kernel is required for a solution of the congruence subgroup problem and also for the theory of automorphic forms of fractional weights. Whenever the congruence subgroup kernel is central, it is isomorphic to the dual of the metaplectic kernel. Thus the computation of the metaplectic kernel leads to a precise solution of the congruence subgroup problem.
In 1987 I gave a formula for the volume of the quotient of a semi-simple group by any principal S-arithmatic subgroup; see [12]. This formula was used in a subsequent paper [13], written jointly with Armand Borel, to prove the finiteness of the number of S-arithmetic subgroups with covolume bounded by a given number and also the finiteness of number of groups of compact type with a given class number. In another joint work [14] with Borel I investigated the values of an irrational isotropic quadratic form at S-integral points. The questions which we raised in this work have now been settled in the affirmative independently by M. Ratner and G. A. Margulis and G. Tomanov.
I have currently been interested in representation theory of reductive groups over local fields. In the papers [15, 16] written in collaboration with Allen Moy, the existence and associativity of unrefined minimal K-types in admissible representation of these groups have been established. In these papers natural filtrations of parahoric subgroups have been introduced and using a result due to G. Kempf and G. Rousseau in Invariant Theory, a uniform proof of existence of unrefined minimal K-types has been provided for all reductive groups. We have defined a new invariant for an admissible representation called the depth, it is a nonnegative rational number attached to the representation. In [16], we proved that the depth does not change under parabolic induction and Jacquet restrictions. in that paper, we have also studied admissible representations of depth zero.
2. Prasad G, Strong rigidity of Q-rank 1 lattices, Inventiones Math, 21 (1973) 255-86.
3. Prasad G, Unipotent elements and isomorphisms of lattices in semi-simple groups, J Indian math Soc, 37 (1973) 103-24.
4. Prasad G, Discrete subgroups isomorphic to lattices in Lie groups, Amer J math, 98 (1976) 241-61, 18.
5. Prasad G, Lattices in semi-simple groups over local fields , Advances in Math Studies in Algebra and Number Theory (1979) 285-356.
6. Prasad G, Nonvanishing of the first cohomology, Bull Soc math Fr, 105 (1977) 415-18.
7. Prasad G, Strong approximation, Ann Math, 105 (1977) 553-72.
8. Prasad G & Raghunathan M S, On the Kneser-Tits problem: Commentarii Math Helv, 60 (1985) 107-21.
9. Prasad G & Raghunathan M S, On the congruence subgroup problem: Determination of the "metaplectic kernel", Inventiones Math, 71 (1983) 21-42.
10. Prasad G & Raghunathan M S, Topological central extensions of semi-simple groups over local fields, Ann Math, 119 (1984) 143-268.
11. Prasad G & Raghunathan M S, Topological central extensions of SL1(D) Inventiones Math, 92 (1988) 645-689.
12. Prasad G, Volumes of S-arithmetic quotients of semi-simple groups, Publ Math IHES, 69 (1989). 91-117.
13. Borel A & Prasad G, finiteness theorems for discrete subgroups of bounded covoulme in semi-simple groups, Publ Math IHES, 69 (1989) 119-171; Addendum:ibid, 71 (1990), 173-177.
14. Borel A & Prasad G, Values of isotropic quadratic forms at S-integral points, Compositio Math, 83 (1992) 347-372.
15. Moy A & Prasad G, Unrefined minimal K-types for p-adic groups, Inventiones Math , 116 (1994) 393-408.
16. Moy A & Prasad G. Jacquet functors and unrefined minimal K-types,
Commentarii Math Helv , 71(1996), 98-121.
17. Prasad G & Rapinchuk A S, Computation of the metaplectic kernel,
Publ Math IHES, 84(1996), 100 pp.