Browder F. (ed.) Mathematics into 21 century (AMS,1992)(T)(509s).djvu: eKnigu

Browder F. (ed.) Mathematics into 21 century (AMS,1992)(T)(509s).djvu

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Date Nov 23, 2004

A key feature of arithmetic groups is that they come in families, defined
by congruence conditions...
To get a feel for the structure of Г, consider the case of
and its subgroup Г}, of integral matrices, which is isomorphic to Z...
The previous paragraph suggests that GR and the Go are on an essentially
equal footing as far as £?{G, Gq) is concerned...
The unitary dual of GA is identifiable to
the (restricted) product of the unitary duals of the local factors Gn , via
a tensor product construction [Flat]...
His pro-
proposal, which has gone through several refinements [Arth2, 3, Borl5, Lusz5],
and may well go through more, can be thought of as a nonabelian general-
generalization of the local reciprocity law of local classfield theory [Lang4, Weil2,
Lgld2]...
This is the true "Langlands classification."
Theorem 3.6.4.5 is a bowdlerized version, expressed solely in terms of the
structure of GR, with reference to &(GR) expunged...
This correspondence has been
studied globally (i.e., for automorphic forms) as well as locally (for admissible
representations) and has resulted, among other things, in the establishment
of some new cases of the Artin conjecture [Lgld5, Tunn2]...
We have given perhaps the most basic example in quantum mechanics, the
harmonic oscillator, in §3.1...
Differential equations, especially nonlinear differential equations, were the
context for some of Lie's original investigations [Hawk, LiEn], and certain
infinite-dimensional Lie algebras were studied by Cartan [Crtn5, GuSt2] in
connection with variational problems...
In the
past decade, there has been a large amount of work generalizing this to the
case of semisimple symmetric spaces (cf...
An easy argument convinces one that T is ergodic if and only if the only
^-invariant functions in L (X, ц) are the constant functions...
The Gelfand-
Fomin argument went through several stages of generalization until today it
can be used to show that nearly any measurable dynamical system coming
from a homogeneous space will be ergodic, unless it fails to be for obvious
reasons [HoMo, Moor2, Zimml]...
In two dimensions, i.e., if dimM = 2, all Hamiltonian systems are com-
completely integrable, but in dimensions greater than two, complete integrabil-
ity is very special, and the discovery of completely integrable systems is an
interesting event...
Understanding the Toda lattice involves looking at this action
from several points of view...
Precisely, under the
covering К —> AdK(y0) by the map k{ —> A.dfcj(y0), the flow D.3.33) on
Ad K(y0) is identified to right translation on B0+\SLn(R) by ехр(Гу/(у0)) •
To tighten the connection made by Proposition 4.3.37, we need another
observation about v/ f°r Ad .K-invariant /...
But this condition is far too restrictive, and hardly ever
holds when V is infinite dimensional...
Thus, in particular, in the context of this section, we
have an action of ^(Lie(G)) on F°° ...
If we let h vary in a one-parameter group and
differentiate at the origin, we may conclude that
(A...
Let H С G be a closed subgroup, and let с be a quasi-isometric repre-
representation of Я on a Banach space U...
A CENTURY OF LIE THEORY 289
Let a be a split Cartan subalgebra of g, as in (A.2.2.1), and consider the
adjoint action of a on g...
Now consider g e G, and let g = кап be its Iwasawa decomposition as
an element of GLn(R)...
Osborne, The n-cohomology of representations with an infinitesi-
infinitesimal character, Compositio Math...