Symmetry of geometry induces symmetry of function spaces through the regular representation. In turn, it provides a group theoretic approach to global analysis such as the classical Fourier series expansion or more generally the spherical harmonics expansions where the "symmetry" is abelian or compact groups. In this talk, we address some basic questions about the global analysis on manifolds X acted algebraically by highly non-commutative groups $G$ such as $\mathit{SL}(n, \mathbb{R})$.

Problem A. Does the group G control "sufficiently" the space of function on $X$?

Problem B. What can we say about "spectrum" for $L^2(X)$?

We plan to discuss some recent progress with emphasis on "multiplicity" for Problem A and "decay of matrix coefficients" for Problem B.

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© Toshiyuki Kobayashi