Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices

By François Maucourant. 2005

62 citations recorded by the source corpus.

We show that Haar measures of connected semisimple groups, embedded via a representation into a matrix space, have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled. This is still true if the Haar measure of the semisimple group is replaced by the Haar measure of a irreducible lattice of the group, and the asymptotic measure is the same. In the case of an almost simple group of rank greater than 2, a remainder term is also obtained. This extends and makes precise anterior results of Duke, Rudnick and Sarnak, and Eskin-McMullen in the case of a group variety.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0c1841aba150d0f497c9352dd1feed5db06df610

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Citations in the corpus

Counting lattice points (2009, background): “(2) via mixing of 1-parameter flows, or equivalently decay of matrix coe‰cients for semisimple groups [1], [2], [6], [27], [37], [59], [60], [61], [62], [65], [66],”
Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds (2008, unknown): “Passing from the above bounds of (6.6) for K -ﬁnite vectors to those for smooth vectors has been detailed in [28, proof of Thm. 6].”
Manin's and Peyre's conjectures on rational points and adelic mixing (2006, unknown): “It is interesting to compare this result with the distribution of the integral points G(Z) of bounded height in the Satake compactification of G(R) where the limiting distribution is supported on the boundary (see [30] and [39] for more details).”
The Ergodic Theory of Lattice Subgroups (2006, methodology): “That is the case also for 3 ≤ n ≤ 6. The case n = 2 is not addressed in [ Ma ].”
Prime and almost prime integral points on principal homogeneous spaces (2009, methodology): “The exponent a has the following simple algebraic description [G-W][ Ma ].”
Igusa integrals and volume asymptotics in analytic and adelic geometry (2009, background): “It is also explained in [33] that the dual face of F is contained in the positive Weyl chamber C (Lemma 2.”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “, when δ = n− 1, both the effective counting and applications to an affine sieve have been extensively studied (see [16], [17], [4], [46], [21], [42],[48], [20], etc.”