International Congress of Mathematicians

762 Semyon Alesker2.3. Valuations invariant under a groupLet G be a subgroup of GL(V). Yet us denote by Val G (V) the space of G-invariant translations invariant continuous valuations. From the results of [2] and[4] follows the following result.2.3.1 Theorem. Let G be a compact subgroup ofGL(V) acting transitively on theunit sphere. Then Val G (V) is a finite dimensional graded subalgebra of (Val(V)) sm .It satisfies the Poincaré duality, and if —Id £ G it satisfies the hard Lefschetztheorem.It turns out that Val G (V) can be described explicitly (as a vector space) forG = SO(n), 0(n), and U(n). In the first two cases it is the classical theorem ofHadwiger [10], the last case is new (see [4]). In order to state these results we haveto introduce first sufficiently many examples.Let 0 be a compact domain in a Euclidean space V with a smooth boundary90. Yet n = dim V. For any point s £ 90 let ki(s),..., fc n _i (s) denote the principalcurvatures at s. For 0 < i < n — 1 defineVi(Q) := - (n T 1 ) f {k h ,..., *,•„_!_, }da,n\n-l-ijwhere {kj x ,..., kj n _ 1 _ i } denotes the (n — l — z)-th elementary symmetric polynomialin the principal curvatures, da is the measure induced on 90 by the Euclideanstructure. It is well known that V» (uniquely) extends by continuity in the Hausdorffmetric to /C(V). Define also F„(0) := vol (Si). Note that Vo is proportional to theEuler characteristic x- It ' 1S weu known that Vo, Vi,..., V n belong to Val°^(V).It is easy to see that \fi is homogeneous of degree k. The famous result of Hadwigersays2.3.2 Theorem (Hadwiger, [10]). LetV be n-dimensional Euclidean space. Thevaluations Vn,Vi,.. .,V n form a basis of Val s °( n fiV)(= Val°( n fiVj).Now let us describe unitarily invariant valuations on a Hermitian space. LetW be a Hermitian space, i.e. a complex vector space equipped with a Hermitianscalar product. Let m := dim^W (thus dim^W = 2m). For every non-negativeintegers p and k such that 2p < k < 2m let us introduce the following valuations:ThenUk, P £J dQUk,p(K)= / V k - 2p (KC\E)-dE.JE^AGr m - pVal" {m) (W).2.3.3 Theorem ([4]). Let W be a Hermitian vector space of complex dimensionm. The valuations Uk, p with 0 < p < """' % m ~ ' form a basis of the spaceVal? m \W).It turns out that the proof of this theorem is highly indirect, and it useseverything known about even translation invariant continuous valuations including