Some contributions to mathematics
Prasad's early work was on discrete subgroups of real and p-adic semi-simple groups. He proved the "strong rigidity" of lattices in real semi-simple groups of rank 1 and also of lattices in p-adic groups, see [1] and [2]. He then tackled group-theoretic and arithmetic questions on semi-simple algebraic groups. He proved the "strong approximation" property for simply connected semi-simple groups over global function fields [3]. Prasad determined the topological central extensions of these groups and computed the "metaplectic kernel" for isotropic groups in collaboration with M. S. Raghunathan, see [11], [12] and [10]. Prasad and Raghunathan have also obtained results on the Kneser-Tits problem, [13]. Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semi-simple groups, see [14].
In 1987, Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups, [4]. Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel and Gopal Prasad proved several finiteness theorems about arithmetic groups, [6]. The volume formula, together with number-theoretic and Bruhat-Tits theoretic considerations led to a classification, by Gopal Prasad and Sai-Kee Yeung, of fake projective planes (in the theory of smooth projective complex surfaces) into 28 non-empty classes [21] (see also [22] and [23]). This classification, together with computations by Donald Cartwright and Tim Steger, has led to a complete list of fake projective planes. This list consists of exactly 50 fake projective planes, up to isometry (distributed among the 28 classes). This work was the subject of a talk in the Bourbaki seminar.
Prasad has worked on the representation theory of reductive p-adic groups with Allen Moy. The filtrations of parahoric subgroups, referred to as the "Moy-Prasad filtration", is widely used in representation theory and harmonic analysis. Moy and Prasad used these filtrations and Bruhat–Tits theory to prove the existence of "unrefined minimal K-types", to define the notion of "depth" of an irreducible admissible representation and to give a classification of representations of depth zero, see [8] and [9]. The results and techniques introduced in these two papers [8],[9] enabled a series of important developments in the field.
In collaboration with Andrei Rapinchuk, Prasad has studied Zariski-dense subgroups of semi-simple groups and proved the existence in such a subgroup of regular semi-simple elements with many desirable properties, [15], [16]. These elements have been used in the investigation of geometric and ergodic theoretic questions. Prasad and Rapinchuk introduced a new notion of "weak-commensurability" of arithmetic subgroups and determined "weak- commensurability classes" of arithmetic groups in a given semi-simple group. They used their results on weak-commensurability to obtain results on length-commensurable and isospectral arithmetic locally symmetric spaces, see [17], [18] and [19].
Together with Jiu-Kang Yu, Prasad has studied the fixed point set under the action of a finite group of automorphisms of a reductive p-adic group G on the Bruhat-Building of G, [24]. In another joint work, that has been used in the geometric Langlands program, Prasad and Yu determined all the quasi-reductive group schemes over a discrete valuation ring (DVR), [25].
In collaboration with Brian Conrad and Ofer Gabber, Prasad has studied the structure of pseudo-reductive groups, and also provided proofs of the conjugacy theorems for general smooth connected linear algebraic groups, announced without detailed proofs by Armand Borel and Jacques Tits; their research monograph [26] contains all this. A second monograph [27] contains a complete classification of pseudo-reductive groups, including a Tits-style classification and also many interesting examples. The classification of pseudo-reductive groups already has many applications. There was a Bourbaki seminar in March 2010 on the work of Tits, Conrad-Gabber-Prasad on pseudo-reductive groups.
Prasad has developed new methods for unramified and tamely ramified descents in Bruhat-Tits theory [28][29]. Together with Tasho Kaletha, he has recently written a book [30] on Bruhat-Tits theory which contains new proofs of several results.
Related Research Articles
Jacques Tits was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
In mathematics, a building is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings plays a role in the study of p-adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups.
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory.
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and semisimple algebraic groups are reductive.
Grigory Aleksandrovich Margulis is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978, a Wolf Prize in Mathematics in 2005, and an Abel Prize in 2020, becoming the fifth mathematician to receive the three prizes. In 1991, he joined the faculty of Yale University, where he is currently the Erastus L. De Forest Professor of Mathematics.
In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were introduced by the mathematician Jacques Tits, and are also sometimes known as Tits systems.
In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by Mostow (1968) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad (1973) in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm. Besson, Courtois & Gallot (1996) gave the simplest available proof.
In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k.
In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflections to –1.
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.
In mathematics, a fake projective plane is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.
In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer.
In algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a nonarchimedean local field that is analogous to a Borel subgroup of an algebraic group. A parahoric subgroup is a proper subgroup that is a finite union of double cosets of an Iwahori subgroup, so is analogous to a parabolic subgroup of an algebraic group. Iwahori subgroups are named after Nagayoshi Iwahori, and "parahoric" is a portmanteau of "parabolic" and "Iwahori". Iwahori & Matsumoto (1965) studied Iwahori subgroups for Chevalley groups over p-adic fields, and Bruhat & Tits (1972) extended their work to more general groups.
In mathematics, a pseudo-reductive group over a field k is a smooth connected affine algebraic group defined over k whose k-unipotent radical is trivial. Over perfect fields these are the same as (connected) reductive groups, but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive. A pseudo-reductive k-group need not be reductive. Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic.
Howard Garland is an American mathematician, who works on algebraic groups, Lie algebras, and infinite-dimensional algebras.
Moshe Jarden is an Israeli mathematician, specialist in field arithmetic.
Shahar Mozes is an Israeli mathematician.
In mathematics, the Moy–Prasad filtration is a family of filtrations of p-adic reductive groups and their Lie algebras, named after Allen Moy and Gopal Prasad. The family is parameterized by the Bruhat–Tits building; that is, each point of the building gives a different filtration. Alternatively, since the initial term in each filtration at a point of the building is the parahoric subgroup for that point, the Moy–Prasad filtration can be viewed as a filtration of a parahoric subgroup of a reductive group.
References
[1]. Strong rigidity of Q-rank 1 lattices, Inventiones Math. 21(1973), 255–286.
[2]. Lattices in semi-simple groups over local fields, Adv.in Math. Studies in Algebra and Number Theory, 1979, 285–356.
[3]. Strong approximation for semi-simple groups over function fields, Annals of Mathematics 105(1977), 553–572.
[4]. Volumes of S-arithmetic quotients of semi-simple groups, Publ.Math.IHES 69(1989), 91–117.
[5]. Semi-simple groups and arithmetic subgroups, Proc.Int.Congress of Math., Kyoto, 1990, Vol. II, 821–832.
[6]. Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ.Math.IHES 69(1989), 119–171; Addendum: ibid, 71(1990); with A.Borel.
[7]. Values of isotropic quadratic forms at S-integral points, Compositio Mathematica, 83 (1992), 347–372; with A.Borel.
[8]. Unrefined minimal K-types for p-adic groups, Inventiones Math. 116(1994), 393–408; with Allen Moy.
[9]. Jacquet functors and unrefined minimal K-types, Commentarii Math.Helv. 71(1996), 98–121; with Allen Moy.
[10]. On the congruence subgroup problem: Determination of the "Metaplectic Kernel", Inventiones Math. 71(1983), 21–42; with M.S.Raghunathan.
[11]. Topological central extensions of semi-simple groups over local fields, Annals of Mathematics 119(1984), 143–268; with M.S.Raghunathan.
[12]. Topological central extensions of SL_1(D), Inventiones Math. 92(1988), 645–689; with M.S.Raghunathan.
[13]. On the Kneser-Tits problem, Commentarii Math.Helv. 60(1985), 107–121; with M.S.Raghunathan.
[14]. Computation of the metaplectic kernel, Publ.Math.IHES 84(1996), 91–187; with A.S.Rapinchuk.
[15]. Existence of irreducible R-regular elements in Zariski-dense subgroups, Math.Res.Letters 10(2003), 21–32; with A.S.Rapinchuk.
[16]. Zariski-dense subgroups and transcendental number theory, Math.Res.Letters 12(2005), 239–249; with A.S.Rapinchuk.
[17]. Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ.Math.IHES 109(2009), 113–184; with A.S.Rapinchuk.
[18]. Local-global principles for embedding of fields with involution into simple algebras with involution, Commentarii Math.Helv. 85(2010), 583–645; with A.S.Rapinchuk.
[19]. On the fields generated by the lengths of closed geodesics in locally symmetric spaces, preprint; with A.S.Rapinchuk.
[20]. Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre, In "Collected papers of John Milnor", vol.V, AMS (2010), 307–325; with A.S.Rapinchuk.
[21]. Fake projective planes, Inventiones Math. 168(2007), 321–370, "Addendum", ibid, 182(2010), 213–227; with Sai-Kee Yeung.
[22]. Arithmetic fake projective spaces and arithmetic fake Grassmannians, Amer.J.Math. 131(2009), 379–407; with Sai-Kee Yeung.
[23]. Nonexistence of arithmetic fake compact hermitian symmetric spaces of type other than A_n, n<5, J.Math.Soc.Japan; with Sai-Kee Yeung.
[24]. On finite group actions on reductive groups and buildings, Inventiones Math. 147(2002), 545–560; with Jiu-Kang Yu.
[25]. On quasi-reductive group schemes, J.Alg.Geom. 15(2006), 507–549; with Jiu-Kang Yu.
[26]. Pseudo-reductive groups, second edition, New Mathematical Monographs #26, xxiv+665 pages, Cambridge University Press, 2015; with Brian Conrad and Ofer Gabber.
[27]. Classification of Pseudo-reductive groups, Annals of Mathematics Studies #191, 245 pages, Princeton University Press, 2015; with Brian Conrad.
[28]. A new approach to unramified descent in Bruhat-Tits theory, Amer. J. Math. vol. 142 #1 (2020), 215–253.
[29]. Finite group actions on reductive groups and buildings and tamely-ramified descent in Bruhat-Tits theory, Amer. J. Math. vol. 142 #4 (2020), 1239–1267.
[30]. Bruhat--Tits theory: a new approach, Cambridge University Press, UK, 2022; with Tasho Kaletha.
