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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)].
Reviewer: H.Rindler (Wien)

##### MSC:
 11G99 Arithmetic algebraic geometry (Diophantine geometry) 22E46 Semisimple Lie groups and their representations 53C35 Differential geometry of symmetric spaces
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