A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

Parabolic contractions of semisimple Lie algebras and their invariants

Let $G$ be a connected semisimple algebraic group with Lie algebra $\mathfrak{g }$ and $P$ a parabolic subgroup of $G$ with $\mathrm{Lie\, }P=\mathfrak{p }$ . The parabolic contraction $\mathfrak{q }$ of $\mathfrak{g }$ is the semi-direct product of $\mathfrak{p }$ and a $\mathfrak{p }$ -module $\mathfrak{g }/\mathfrak{p }$ regarded as an abelian ideal. We are interested in the polynomial invariants of the adjoint and coadjoint representations of $\mathfrak{q }$ . In the adjoint case, the algebra of invariants is easily described and it turns out to be a graded polynomial algebra. The coadjoint case is more complicated. Here we found a connection between symmetric invariants of $\mathfrak{q }$ and symmetric invariants of centralisers $\mathfrak{g }_e\subset \mathfrak{g }$ , where $e\in \mathfrak{g }$ is a Richardson element with polarisation $\mathfrak{p }$ . Using this connection and results of Panyushev et al. (J Algebra 313:343–391, 2007), we prove that the algebra of symmetric invariants of $\mathfrak{q }$ is free for all parabolic subalgebras in types $\mathbf A$ and $\mathbf C$ and some parabolics in type $\mathbf B$ . This technique also applies to the minimal parabolic subalgebras in all types. For $\mathfrak{p }=\mathfrak{b }$ , a Borel subalgebra of $\mathfrak{g }$ , one gets a contraction of $\mathfrak{g }$ recently introduced by Feigin (Selecta Math 18:513–537, 2012) and studied from invariant-theoretic point of view in our previous paper (Panyushev and Yakimova in Ann Inst Fourier 62(6):2053–2068, 2012).     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

On the Left and Right Brylinski-Kostant Filtrations

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ a Borel subalgebra, and $\mathfrak{h}\subset\mathfrak{b}$ a Cartan subalgebra. Let V be a finite dimensional simple $U(\mathfrak{g})$ module. Based on a principal s-triple (e,h,f) and following work of Kostant, Brylinski (J Amer Math Soc 2(3):517–533, 1989) defined a filtration $\mathcal{F}_e$ for all weight subspaces V _μ of V and calculated the dimensions of the graded subspaces for μ dominant. In Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) these dimensions were calculated for all μ. Let δM(0) be the finite dual of the Verma module of highest weight 0. It identifies with the global functions on a Weyl group translate of the open Bruhat cell and so inherits a natural degree filtration. On the other hand, up to an appropriate shift of weights, there is a unique $U(\mathfrak{b})$ module embedding of V into δM(0) and so the degree filtration induces a further filtration $\mathcal{F}$ on each weight subspace V _μ . A casual reading of Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) might lead one to believe that $\mathcal{F}$ and $\mathcal{F}_e$ coincide. However this is quite false. Rather one should view $\mathcal{F}_e$ as coming from a left action of $U(\mathfrak{n})$ and then there is a second (Brylinski-Kostant) filtration $\mathcal{F}'_e$ coming from a right action. It is $\mathcal{F}'_e$ which may coincide with $\mathcal{F}$ . In this paper the above claim is made precise. First it is noted that $\mathcal{F}$ is itself not canonical, but depends on a choice of variables. Then it is shown that a particular choice can be made to ensure that $\mathcal{F}=\mathcal{F}'_e$ . An explicit form for the unique left $U(\mathfrak{b})$ module embedding $V\hookrightarrow\delta M(0)$ is given using the right action of $U(\mathfrak{n})$ . This is used to give a purely algebraic proof of Brylinski’s main result in Brylinski (J Amer Math Soc 2(3):517–533, 1989) which is much simpler than Joseph et al. (J Amer Math Soc 13(4):945–970, 2000). It is noted that the dimensions of the graded subspaces of $\rm{gr}_{\mathcal{F}_e} V_{\!\mu}$ and $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ can be very different. Their interrelation may involve the Kashiwara involution. Indeed a combinatorial formula for multiplicities in tensor products involving crystal bases and the Kashiwara involution is recovered. Though the dimensions for the graded subspaces of $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ are determined by polynomial degree, their values remain unknown.     

On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.     

Complement of forms

The notions of the parallel sum, the parallel difference, and the complement of two nonnegative sesquilinear forms were introduced and studied by Hassi, Sebestyé and de Snoo in Hassi et al. (Oper Theory Adv Appl 198:211–227, 2010) and Hassi et al. (J Funct Anal 257(12):3858–3894, 2009). In this paper we continue these investigations. The Galois correspondence induced by the map ${\mathfrak{m} \mapsto \mathfrak{m}_\mathfrak{t}}$ (where ${\mathfrak{m}_\mathfrak{t}}$ denotes the ${\mathfrak{t}}$ -complement of ${\mathfrak{m}}$ ) is also studied. Inspired by the work of Eriksson and Leutwiler Eriksson and Leutwiler (Math Ann 274:301–317, 1986), we introduce the notion of quasi-unit for nonnegative sesquilinear forms. The quasi-units are characterized by means of the complement and the disjoint part. It is also shown that the ${{\mathfrak{t}}}$ -quasi-units coincide with the extreme points of the convex set ${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$ .     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

Graphs, partitions and classes of groups

If ${\mathfrak X}$ is a class of groups, Delizia et?al. (Bull Austral Math Soc 75:313–320, 2007) call a group G ${\mathfrak X}$ -transitive (or an ${\mathfrak XT}$ -group) if whenever ${\langle a,b\rangle}$ and ${\langle b,c\rangle}$ are in ${\mathfrak X} \langle a,c\rangle$ is also in ${\mathfrak X}$ ( ${a,b,c\in G}$ ). The structure of ${\mathfrak XT}$ -groups has been investigated for a number of classes of groups, by Delizia, Moravec and Nicotera and others. A graph can be associated with a group in many ways. Delizia, Moravec and Nicotera introduce a graph which is a generalisation of the commuting graph of a group, but do not make use of the graph. We will use the properties of the graph to investigate further classes of groups and to obtain more detailed structural information.     

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.     

Supports of (\mathfrak{g},\mathfrak{k})-modules of finite type

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{k}$ be a reductive subalgebra in $\mathfrak{g}$ . We say that a $\mathfrak{g}$ -module M is a $(\mathfrak{g},\mathfrak{k})$ -module if M, considered as a $\mathfrak{k}$ -module, is a direct sum of finite-dimensional $\mathfrak{k}$ -modules. We say that a $(\mathfrak{g},\mathfrak{k})$ -module M is of finite type if all $\mathfrak{k}$ -isotopic components of M are finite-dimensional. In this paper we prove that any simple $(\mathfrak{g},\mathfrak{k})$ -module of finite type is holonomic. A simple $\mathfrak{g}$ -module M is associated with the invariants V(M), V(LocM), and L(M) reflecting the ??directions of growth of M.?? We also prove that for a given pair $(\mathfrak{g},\mathfrak{k})$ the set of possible invariants is finite.     

On changing highest weight theories for finite W-algebras

A highest weight theory for a finite W-algebra $U(\mathfrak{g},e)$ was developed in Brundan et al. (Int. Math. Res. Not. 15:rnn051, 2008). This leads to a strategy for classifying the irreducible finite dimensional $U(\mathfrak{g},e)$ -modules. The highest weight theory depends on the choice of a parabolic subalgebra of $\mathfrak{g}$ leading to different parameterizations of the finite dimensional irreducible $U(\mathfrak{g},e)$ -modules. We explain how to construct an isomorphism preserving bijection between the parameterizing sets for different choices of parabolic subalgebra when $\mathfrak{g}$ is of type A, or when $\mathfrak{g}$ is of types C or D and e is an even multiplicity nilpotent element.     

Varieties Generated by Wreath Products of Abelian Groups

This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups $ \mathfrak{X} $ Wr $ \mathfrak{Y} $ = { X Wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } and $ \mathfrak{X} $ wr $ \mathfrak{Y} $ = {X wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } generate the product variety $ \mathfrak{X} $ var ( $ \mathfrak{Y} $ ).     

On Deformations of Lie Algebroids

For any Lie algebroid A, its 1-jet bundle ${\mathfrak{J} A}$ is a Lie algebroid naturally and there is a representation ${\pi:\mathfrak{J} A\longrightarrow\mathfrak{D} A}$ . Denote by ${{\rm d}_{\mathfrak{J}}}$ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A,A)_{{\mathfrak{D}} A}),{\rm d}_{\mathfrak{J}})}$ of the cochain complex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A, A)),{\rm d}_\mathfrak{J})}$ .     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

The geometric meaning of Zhelobenko operators

Let $ \mathfrak{g} $ be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, $ \mathfrak{b} $ a Borel subalgebra of $ \mathfrak{g} $ , $ \mathfrak{h}\subset \mathfrak{b} $ the Cartan sublagebra, and N ? G the unipotent subgroup corresponding to the nilradical $ \mathfrak{n}\subset \mathfrak{b} $ . We show that the explicit formula for the extremal projection operator for $ \mathfrak{g} $ obtained by Asherova, Smirnov, and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence $ N\times \mathfrak{h}\to \mathfrak{b} $ given by the restriction of the adjoint action. Simple geometric proofs of formulas for the “classical” counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.     

On the generalization of Faltings’ Annihilator Theorem

Let R be a commutative Noetherian ring, and let n be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever R is a homomorphic image of a Noetherian Gorenstein ring, then the invariants ${\inf\{i \in \mathbb{N}_0|\, \rm{dim\, Supp}(\mathfrak{b}^t H_{\mathfrak{a}}^i(M)) \geq n\, \rm{for\, all}\, t \in \mathbb{N}_0\}}$ and ${\inf\{\lambda_{\mathfrak{a} R_{\mathfrak{p}}}^{\mathfrak{b} R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\, \mathfrak{p} \in {\rm Spec} \, R\, \rm{and\, dim}\, R/ \mathfrak{p} \geq n\}}$ are equal, for every finitely generated R-module M and for all ideals ${\mathfrak{a}, \mathfrak{b}}$ of R with ${\mathfrak{b}\subseteq \mathfrak{a}}$ . This generalizes Faltings’ Annihilator Theorem (see [6]).     

Morrey-type regularity of solutions to parabolic problems with discontinuous data

We consider regular oblique derivative problem in cylinder Q _T ?=????× (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203?C220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$ , ${p\in(1,\infty)}$ . Our aim is to show that the solution belongs to the generalized Sobolev?CMorrey space ${W^{2,1}_{p,\omega}(Q_T)}$ , when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$ , ${p\in (1, \infty)}$ , ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$ . For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy?CDirichlet problem.     

Erratum to: Algèbres de lie quasi-réductives

In Corollary 12(ii) and Theorem 13(v) of [1] we omitted the hypothesis dim $ \mathfrak{z}\leq 1 $ . Moreover, in some places the symbol $ \mathbb{K} $ must be replaced by the symbol $ {{\mathbb{K}}^{\times }} $ .     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .