- AUTOMORPHIC FORMS AND THE REGULAR REPRESENTATION ON r\G 31
supported functions by the rule:
(83) 7r(a)v = [ a(x)n(x)vdx (v EV)The subspace spanned by the 7r(a)v, where v runs through Vanda through C'g°(G)
is dense, as is easily seen by using Dirac sequences ([4], 3.4). Call it V^00. It consists
of C^00 vectors (i.e. such that g 1-t n(g )v is a smooth map), and it used to be called
the Garding space. However, a special terminology has become superfluous because
a theorem of Dixmier-Malliavin ([11]) implies that these are all differentiable vec-
tors, but we shall not need this fact. V^00 is a G-module upon which g operates
naturally ([4], 3.8), hence V^00 is also a U(g)-module.
9.2. The operation defined by (83) above extends to compactly supported mea-
sures. In particular, V can be viewed as a K-module, and as such is completely
reducible and for A E k, the transformation defined by
(84) n(e,)V = d(..) [ X,(k)n(k).vdk
(see Proposition 3.6 for the notation) is a projector of V onto the isotypic
subspace V,* spanned by the irreducible K-modules isomorphic to the module ..*
contragredient to.>-. Let Vi< be the algebraic direct sum of the Vi. It consists of the
K-finite vectors and is dense in V. Any a E C'g°(G) which is K-invariant leaves the
V,\ stable; using a Dirac sequence of such functions, one sees that VJ( = V^00 n Vx
is dense in V and that V^00 n V,\ is dense in V,. In particular, V,\ c V^00 if V,\ is
finite-dimensional.
9.3. The actions of U(g) and Kon VJ( satisfy the conditions
(85) 7r(k)n(X)v = n(Ad(k)X)n(k)v (k EK, XE U(g), v EV)
which implies that
(1) U(g) leaves VJ( stable.
(2) If W is a K-stable finite dimensional subspace of V, then the representa-
tion of K on W is differentiable, and has 7rlt as its differential.Conditions (1) and (2) define the notion of a (g, K)-module. It is admissible, or a
Harish-Chandra module, if each V,\ is finite dimensional, in which case Vx c V^00.
V/f can be viewed as a module over a so called Hecke algebra
H = H( G, K), which is the convolution algebra of distributions on G with support
in K, and is isomorphic toU(g)©u(e)AK, where Ax is the algebra of finite measures
on K, and U(g) is viewed as the algebra of distributions with support {1}, but
this interpretation plays no role in this course. (However, it does in the adelic
framework.)
9.4. An element v E V^00 is Z(g)-finite if Z(g).v is finite dimensional, hence if v
is annihilated by an ideal of finite codimension of Z(g). The theorem of Harish-
Chandra used in Section 3.2 states that if W C V 1 ( is a finite-dimensional space
of Z(g)-and K-finite vectors, and U a neighborhood of 1 in G, then there exists
a E C'g°(U)K such that n(a) is the identity on W.
9.5. Assume now (n, V) to be unitary and irreducible. By a theorem of Harish-
Chandra ([4], 5.25) it is admissible, hence Vx = VJ(. By a version of Schur's