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{{Short description|Group representation via algebra automorphisms}} {{distinguish|algebra representation}} ...

...ion]] of a [[group (mathematics)|group]] or a [[Lie algebra representation|Lie algebra]] on a [[symplectic vector space]] (''V'', ''ω'') which preserves t ...'''F''' is the [[field (mathematics)|field]] of scalars. A representation of a group ''G'' preserves ''ω'' if ...

...<math>(\mathfrak{g},K)</math>-module, then its Harish-Chandra module is a representation with desirable factorization properties. ...dra module'' of <math>\pi</math> is the subspace ''X'' of ''V'' consisting of the [[K-finite]] smooth vectors in ''V''. This means that ''X'' includes ex ...

In geometry, an '''abelian Lie group''' is a [[Lie group]] that is an [[abelian group]]. ...nnected compact complex Lie group is a [[complex torus]]; i.e., a quotient of <math>\mathbb{\Complex}^n</math> by a lattice. ...

...ion|Linear representation of a group on the tangent space to a fixed point of the group.}} ...on''' is a natural [[linear representation]] of a [[Lie group]], that is [[Lie group action|acting]] on a manifold, on the [[tangent space]] to a fixed po ...

...arish-Chandra transform''' is a linear map from functions on a [[reductive Lie group]] to functions on a [[Borel subgroup|parabolic subgroup]]. It was int The Harish-Chandra transform ''f''^''P'' of a function ''f'' on the group ''G'' is given by ...

...ch are invariant under the [[group action|action]] of a Lie group in terms of functions on a [[Cartan subalgebra]]. | ''G'' || complex connected [[semisimple Lie group]] || SL_''n'', the [[special linear group]] ...

{{Short description|Formula in representation theory}} ...imple complex [[Lie algebra]] in a [[tensor product]] of two [[irreducible representation]]s. ...

...e case of an [[Engel group]]. The Engel identity is the defining condition of an [[Engel group]]. ...ng <math>L</math>. The Lie ring <math>L</math> is defined to be an n-Engel Lie ring if and only if ...

...of ''G'' and ''K'' is a [[maximal compact subgroup]] of ''G''.<ref>Page 12 of {{harvnb|Doran|Varadarajan|2000}}</ref> ...sentation]] of ''K'' (without regard to the [[topological space|topology]] of ''K'') satisfying the following three conditions ...

...H'' of a [[finite group]] ''G''. It is a special case of a [[Hecke algebra of a locally compact group]]. ...]] zero, ''G'' a finite [[group (mathematics)|group]] and ''H'' a subgroup of ''G''. Let <math>F[G]</math> denote the ...

...lass]] <math>\lambda_V</math> of a complex [[Representation of a Lie group|representation]] <math>V</math>. The term "cannibalistic" for these classes was introduced *{{citation|first=J. Frank |last=Adams|authorlink=Frank Adams| title=On the groups <math>J(X)</math>. II.|journal=[[Topology (journal)|Topology]] |volume=3|ye ...

...(discrete subgroup)|lattices]] in Lie groups, often under the name ''field of definition''. == Fuchsian and Kleinian groups == ...

...andra|1976a}}; {{harvtxt|Harish-Chandra|1976b}}</ref> Trombi gave a survey of Harish-Chandra's work on this.<ref>{{harvtxt|Trombi|1989}}</ref> *''x'' is an element of a semisimple group ''G'' ...

{{Short description|Lie algebra all of which elements are semisimple}} ...\lambda = 0</math>, we have that <math>-\lambda y</math> is an eigenvector of <math>\operatorname{ad}_{\mathfrak{h}}(y)</math> with eigenvalue zero, a co ...

...d compact Lie group|irreducible representations]] of a connected compact [[Lie group]] <math>K</math>.<ref name="12.6">{{harvnb|Hall|2015}} Theorem 12.6</ ...n the two results is in the precise notion of "integral" in the definition of a dominant integral element. If <math>K</math> is simply connected, this di ...

{{Lie groups |Semi-simple}} ...e.com/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref> Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center. ...

...''' is a [[special function|special]] spherical function on a [[semisimple Lie group]], studied by {{harvs|txt|last=Harish-Chandra|authorlink=Harish-Chand ...vtxt|Wallach|1988|loc=4.5}} gives a detailed description of the properties of ''Ξ''. ...

...ce of [[tempered representation|tempered distributions]] on a semisimple [[Lie group]]. The definition of the Schwartz space uses [[Harish-Chandra's Ξ function]] and his ''σ'' funct ...

...the [[representation theory]] of [[algebraic group]]s. She is a professor of mathematics at the [[École Polytechnique Fédérale de Lausanne]] in Switzerl ...egon]] in 1985. Her dissertation, ''Certain Embeddings of Simple Algebraic Groups'', was supervised by [[Gary Seitz]].{{r|mg}} As a faculty member at [[Wesle ...