##
**Mixing, counting, and equidistribution in Lie groups.**
*(English)*
Zbl 0798.11025

The authors discuss results on mixing, counting and equidistribution in general affine symmetric spaces \(V=G/H\), \(G\) a connected semisimple Lie group with finite center, \(H\) a closed subgroup which is the fixed point of some (not necessarily Cartan-) involution. Let \(\Gamma \subset G\) be a lattice (i.e. \(\Gamma\) is discrete, and \(\Gamma \backslash G\) has finite volume), that has dense projection to \(G/G'\) for any positive-dimensional normal noncompact Lie subgroup \(G' \subseteq G\) and such that \(\Gamma \cap H\) is a lattice in \(H\). The following mixing theorem for \(X = \Gamma \backslash G\) is a well known consequence of the Howe-Moore theorem on the decay of matrix coefficients of unitary representations, [see R. E. Howe and C. C. Moore, J. Funct. Anal. 32, 72-96 (1979; Zbl 0404.22015)]: \(\int \alpha (xg) \beta (x)dx \to (\int_ x \alpha \int_ x \beta)/m(x)\) as \(g\) tends to infinity. The authors prove that the translates \(Yg\) of the \(H\)-orbit \(Y = (\Gamma \cap H) \backslash H\) become equidistributed on \(X = \Gamma \backslash G\). That is
\[
{1 \over m (Y)} \int_{Yg} f(h)dh \to {1 \over m(X)} \int_ Xf(x)dx
\]
as \(g\) leaves compact subsets of \(H \backslash G\). Finally a counting theorem is shown which contains some of the main results of a paper of W. Duke, Z. Rudnick and P. Sarnak [ibid. 71, No. 1, 143-179 (1993); see the review above]. For a sequence of certain well rounded sets \(B_ n\) it is shown that the cardinality of the number of points of \(\Gamma v\) \((v\) denotes the coset \(H\) in \(G/H)\) which lie in \(B_ n\) grows like the volume of \(B_ n\) asymptotically
\[
| \Gamma v \cap B_ n | \sim {m \bigl( (\Gamma \cap H) \backslash H \bigr) \over m(\Gamma \backslash G)} m(B_ n).
\]
The paper starts with interesting examples concerning the hyperbolic plane which essentially already appeared in the thesis of G. A. Margulis [Funct. Anal. Appl. 4, 335-336 (1969); translation from Funkts. Anal. Prilozh. 3, 89-90 (1969; Zbl 0207.203)]. The geometric intuition of negative curvature transparent in this setting leads to the wave front lemma, which is above all proved for the case \(Sl_ n (\mathbb{R})/SO_ n (\mathbb{R})\), which contain all essential ideas.

In general one has for \(G=HAK\), generalizing the polar decomposition \(KAK\) for a Riemannian symmetric space: For any open neighbourhood \(U\) of the identity in \(G\), there exists an open set \(V \subset G\) such that \(HVg \subset HgU\) for all \(g\) in \(AK\). From the wave front lemma one easily deduces the equidistribution theorem which is then used to prove the counting results.

The authors conclude with some interesting non affine symmetric examples: An equidistribution theorem holds for closed horocycles on hyperbolic surfaces of finite volume which need not be affine symmetric (e.g. \(G = PSl_ 2(\mathbb{R})\), \(H\) a maximal unipotent subgroup). For \(G = Sl(2,\mathbb{C})\), \(H\) the subgroup of real diagonal matrices, \(\Gamma = gSl (2,\mathbb{Z} [i]) g^{-1}\), \(g\) a real matrix chosen to conjugate a hyperbolic element of \(Sl_ 2 (\mathbb{Z})\) into \(H\), then \(\Gamma\) meets \(H\) in a lattice, but \(G/H\) is not affine symmetric, then the analogous equidistribution and counting results do not hold but a modified “counting theorem on average” remains valid. Further progress in this direction is contained in A. Eskin, S. Mozes, and N. Shah [Unipotent flows and counting lattice points on homogeneous spaces (in preparation)].

In general one has for \(G=HAK\), generalizing the polar decomposition \(KAK\) for a Riemannian symmetric space: For any open neighbourhood \(U\) of the identity in \(G\), there exists an open set \(V \subset G\) such that \(HVg \subset HgU\) for all \(g\) in \(AK\). From the wave front lemma one easily deduces the equidistribution theorem which is then used to prove the counting results.

The authors conclude with some interesting non affine symmetric examples: An equidistribution theorem holds for closed horocycles on hyperbolic surfaces of finite volume which need not be affine symmetric (e.g. \(G = PSl_ 2(\mathbb{R})\), \(H\) a maximal unipotent subgroup). For \(G = Sl(2,\mathbb{C})\), \(H\) the subgroup of real diagonal matrices, \(\Gamma = gSl (2,\mathbb{Z} [i]) g^{-1}\), \(g\) a real matrix chosen to conjugate a hyperbolic element of \(Sl_ 2 (\mathbb{Z})\) into \(H\), then \(\Gamma\) meets \(H\) in a lattice, but \(G/H\) is not affine symmetric, then the analogous equidistribution and counting results do not hold but a modified “counting theorem on average” remains valid. Further progress in this direction is contained in A. Eskin, S. Mozes, and N. Shah [Unipotent flows and counting lattice points on homogeneous spaces (in preparation)].

Reviewer: H.Rindler (Wien)

### MSC:

11G99 | Arithmetic algebraic geometry (Diophantine geometry) |

22E46 | Semisimple Lie groups and their representations |

53C35 | Differential geometry of symmetric spaces |

### Keywords:

mixing; counting; equidistribution; affine symmetric spaces; semisimple Lie group; wave front lemma
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\textit{A. Eskin} and \textit{C. McMullen}, Duke Math. J. 71, No. 1, 181--209 (1993; Zbl 0798.11025)

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### References:

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