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Tempered representation

In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space

L2+ε(G)

for any ε > 0.

Formulation edit

This condition, as just given, is slightly weaker than the condition that the matrix coefficients are square-integrable, in other words lie in

L2(G),

which would be the definition of a discrete series representation. If G is a linear semisimple Lie group with a maximal compact subgroup K, an admissible representation ρ of G is tempered if the above condition holds for the K-finite matrix coefficients of ρ.

The definition above is also used for more general groups, such as p-adic Lie groups and finite central extensions of semisimple real algebraic groups. The definition of "tempered representation" makes sense for arbitrary unimodular locally compact groups, but on groups with infinite centers such as infinite central extensions of semisimple Lie groups it does not behave well and is usually replaced by a slightly different definition. More precisely, an irreducible representation is called tempered if it is unitary when restricted to the center Z, and the absolute values of the matrix coefficients are in L2+ε(G/Z).

Tempered representations on semisimple Lie groups were first defined and studied by Harish-Chandra (using a different but equivalent definition), who showed that they are exactly the representations needed for the Plancherel theorem. They were classified by Knapp and Zuckerman, and used by Langlands in the Langlands classification of irreducible representations of a reductive Lie group G in terms of the tempered representations of smaller groups.

History edit

Irreducible tempered representations were identified by Harish-Chandra in his work on harmonic analysis on a semisimple Lie group as those representations that contribute to the Plancherel measure. The original definition of a tempered representation, which has certain technical advantages, is that its Harish-Chandra character should be a "tempered distribution" (see the section about this below). It follows from Harish-Chandra's results that it is equivalent to the more elementary definition given above. Tempered representations also seem to play a fundamental role in the theory of automorphic forms. This connection was probably first realized by Satake (in the context of the Ramanujan-Petersson conjecture) and Robert Langlands and served as a motivation for Langlands to develop his classification scheme for irreducible admissible representations of real and p-adic reductive algebraic groups in terms of the tempered representations of smaller groups. The precise conjectures identifying the place of tempered representations in the automorphic spectrum were formulated later by James Arthur and constitute one of the most actively developing parts of the modern theory of automorphic forms.

Harmonic analysis edit

Tempered representations play an important role in the harmonic analysis on semisimple Lie groups. An irreducible unitary representation of a semisimple Lie group G is tempered if and only if it is in the support of the Plancherel measure of G. In other words, tempered representations are precisely the class of representations of G appearing in the spectral decomposition of L2 functions on the group (while discrete series representations have a stronger property that an individual representation has a positive spectral measure). This stands in contrast with the situation for abelian and more general solvable Lie groups, where a different class of representations is needed to fully account for the spectral decomposition. This can be seen already in the simplest example of the additive group R of the real numbers, for which the matrix elements of the irreducible representations do not fall off to 0 at infinity.

In the Langlands program, tempered representations of real Lie groups are those coming from unitary characters of tori by Langlands functoriality.

Examples edit

  • The Plancherel theorem for a semisimple Lie group involves representations that are not the discrete series. This becomes clear already in the case of the group SL2(R). The principal series representations of SL2(R) are tempered and account for the spectral decomposition of functions supported on the hyperbolic elements of the group. However, they do not occur discretely in the regular representation of SL2(R).
  • The two limit of discrete series representations of SL2(R) are tempered but not discrete series (even though they occur "discretely" in the list of irreducible unitary representations).
  • For non-semisimple Lie groups, representations with matrix coefficients in L2+ε do not always suffice for the Plancherel theorem, as shown by the example of the additive group R of real numbers and the Fourier integral; in fact, all irreducible unitary representations of R contribute to the Plancherel measure, but none of them have matrix coefficients in L2+ε.
  • The complementary series representations of SL2(R) are irreducible unitary representations that are not tempered.
  • The trivial representation of a group G is an irreducible unitary representation that is not tempered unless G is compact.

Classification edit

The irreducible tempered representations of a semisimple Lie group were classified by Knapp and Zuckerman (1976, 1982). In fact they classified a more general class of representations called basic representations. If P=MAN is the Langlands decomposition of a cuspidal parabolic subgroup, then a basic representation is defined to be the parabolically induced representation associated to a limit of discrete series representation of M and a unitary representation of the abelian group A. If the limit of discrete series representation is in fact a discrete series representation, then the basic representation is called an induced discrete series representation. Any irreducible tempered representation is a basic representation, and conversely any basic representation is the sum of a finite number of irreducible tempered representations. More precisely, it is a direct sum of 2r irreducible tempered representations indexed by the characters of an elementary abelian group R of order 2r (called the R-group). Any basic representation, and consequently any irreducible tempered representation, is a summand of an induced discrete series representation. However it is not always possible to represent an irreducible tempered representation as an induced discrete series representation, which is why one considers the more general class of basic representations.

So the irreducible tempered representations are just the irreducible basic representations, and can be classified by listing all basic representations and picking out those that are irreducible, in other words those that have trivial R-group.

Tempered distributions edit

Fix a semisimple Lie group G with maximal compact subgroup K. Harish-Chandra (1966, section 9) defined a distribution on G to be tempered if it is defined on the Schwartz space of G. The Schwartz space is in turn defined to be the space of smooth functions f on G such that for any real r and any function g obtained from f by acting on the left or right by elements of the universal enveloping algebra of the Lie algebra of G, the function

 

is bounded. Here Ξ is a certain spherical function on G, invariant under left and right multiplication by K, and σ is the norm of the log of p, where an element g of G is written as : g=kp for k in K and p in P.

References edit

  • Cowling, M., Haagerup, U., Howe, R. Almost L2 matrix coefficients J. Reine Angew. Math. 387 (1988), 97—110
  • Harish-Chandra (1966), "Discrete series for semisimple Lie groups. II. Explicit determination of the characters", Acta Mathematica, 116 (1): 1–111, doi:10.1007/BF02392813, ISSN 0001-5962, MR 0219666, S2CID 125806386
  • Knapp, Anthony W.; Zuckerman, Gregg (1976), "Classification of irreducible tempered representations of semi-simple Lie groups", Proceedings of the National Academy of Sciences of the United States of America, 73 (7): 2178–2180, Bibcode:1976PNAS...73.2178K, doi:10.1073/pnas.73.7.2178, ISSN 0027-8424, JSTOR 65732, MR 0460545, PMC 430485, PMID 16592331
  • Knapp, Anthony W.; Zuckerman, Gregg J. (1982), "Classification of irreducible tempered representations of semisimple groups. Part I", Annals of Mathematics, Second Series, 116 (2): 389–455, doi:10.2307/2007066, ISSN 0003-486X, JSTOR 2007066, MR 0672840 Knapp, Anthony W.; Zuckerman, Gregg J. (1982), "Classification of irreducible tempered representations of semisimple groups. Part II", Annals of Mathematics, Second Series, 116 (3): 457–501, doi:10.2307/2007019, ISSN 0003-486X, JSTOR 2007019, MR 0672840 Knapp, Anthony W.; Zuckerman, Gregg J. (1984), "Correction", Annals of Mathematics, Second Series, 119 (3): 639, doi:10.2307/2007089, ISSN 0003-486X, JSTOR 2007089, MR 0744867
  • Knapp, Anthony W. Representation Theory of Semisimple Groups: An Overview Based on Examples. ISBN 0-691-09089-0
  • Wallach, Nolan. Real reductive groups. I. Pure and Applied Mathematics, 132. Academic Press, Inc., Boston, MA, 1988. xx+412 pp. ISBN 0-12-732960-9

tempered, representation, mathematics, tempered, representation, linear, semisimple, group, representation, that, basis, whose, matrix, coefficients, space, contents, formulation, history, harmonic, analysis, examples, classification, tempered, distributions, . In mathematics a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space L2 e G for any e gt 0 Contents 1 Formulation 2 History 3 Harmonic analysis 4 Examples 5 Classification 6 Tempered distributions 7 ReferencesFormulation editThis condition as just given is slightly weaker than the condition that the matrix coefficients are square integrable in other words lie in L2 G which would be the definition of a discrete series representation If G is a linear semisimple Lie group with a maximal compact subgroup K an admissible representation r of G is tempered if the above condition holds for the K finite matrix coefficients of r The definition above is also used for more general groups such as p adic Lie groups and finite central extensions of semisimple real algebraic groups The definition of tempered representation makes sense for arbitrary unimodular locally compact groups but on groups with infinite centers such as infinite central extensions of semisimple Lie groups it does not behave well and is usually replaced by a slightly different definition More precisely an irreducible representation is called tempered if it is unitary when restricted to the center Z and the absolute values of the matrix coefficients are in L2 e G Z Tempered representations on semisimple Lie groups were first defined and studied by Harish Chandra using a different but equivalent definition who showed that they are exactly the representations needed for the Plancherel theorem They were classified by Knapp and Zuckerman and used by Langlands in the Langlands classification of irreducible representations of a reductive Lie group G in terms of the tempered representations of smaller groups History editIrreducible tempered representations were identified by Harish Chandra in his work on harmonic analysis on a semisimple Lie group as those representations that contribute to the Plancherel measure The original definition of a tempered representation which has certain technical advantages is that its Harish Chandra character should be a tempered distribution see the section about this below It follows from Harish Chandra s results that it is equivalent to the more elementary definition given above Tempered representations also seem to play a fundamental role in the theory of automorphic forms This connection was probably first realized by Satake in the context of the Ramanujan Petersson conjecture and Robert Langlands and served as a motivation for Langlands to develop his classification scheme for irreducible admissible representations of real and p adic reductive algebraic groups in terms of the tempered representations of smaller groups The precise conjectures identifying the place of tempered representations in the automorphic spectrum were formulated later by James Arthur and constitute one of the most actively developing parts of the modern theory of automorphic forms Harmonic analysis editTempered representations play an important role in the harmonic analysis on semisimple Lie groups An irreducible unitary representation of a semisimple Lie group G is tempered if and only if it is in the support of the Plancherel measure of G In other words tempered representations are precisely the class of representations of G appearing in the spectral decomposition of L2 functions on the group while discrete series representations have a stronger property that an individual representation has a positive spectral measure This stands in contrast with the situation for abelian and more general solvable Lie groups where a different class of representations is needed to fully account for the spectral decomposition This can be seen already in the simplest example of the additive group R of the real numbers for which the matrix elements of the irreducible representations do not fall off to 0 at infinity In the Langlands program tempered representations of real Lie groups are those coming from unitary characters of tori by Langlands functoriality Examples editThe Plancherel theorem for a semisimple Lie group involves representations that are not the discrete series This becomes clear already in the case of the group SL2 R The principal series representations of SL2 R are tempered and account for the spectral decomposition of functions supported on the hyperbolic elements of the group However they do not occur discretely in the regular representation of SL2 R The two limit of discrete series representations of SL2 R are tempered but not discrete series even though they occur discretely in the list of irreducible unitary representations For non semisimple Lie groups representations with matrix coefficients in L2 e do not always suffice for the Plancherel theorem as shown by the example of the additive group R of real numbers and the Fourier integral in fact all irreducible unitary representations of R contribute to the Plancherel measure but none of them have matrix coefficients in L2 e The complementary series representations of SL2 R are irreducible unitary representations that are not tempered The trivial representation of a group G is an irreducible unitary representation that is not tempered unless G is compact Classification editThe irreducible tempered representations of a semisimple Lie group were classified by Knapp and Zuckerman 1976 1982 In fact they classified a more general class of representations called basic representations If P MAN is the Langlands decomposition of a cuspidal parabolic subgroup then a basic representation is defined to be the parabolically induced representation associated to a limit of discrete series representation of M and a unitary representation of the abelian group A If the limit of discrete series representation is in fact a discrete series representation then the basic representation is called an induced discrete series representation Any irreducible tempered representation is a basic representation and conversely any basic representation is the sum of a finite number of irreducible tempered representations More precisely it is a direct sum of 2r irreducible tempered representations indexed by the characters of an elementary abelian group R of order 2r called the R group Any basic representation and consequently any irreducible tempered representation is a summand of an induced discrete series representation However it is not always possible to represent an irreducible tempered representation as an induced discrete series representation which is why one considers the more general class of basic representations So the irreducible tempered representations are just the irreducible basic representations and can be classified by listing all basic representations and picking out those that are irreducible in other words those that have trivial R group Tempered distributions editFix a semisimple Lie group G with maximal compact subgroup K Harish Chandra 1966 section 9 defined a distribution on G to be tempered if it is defined on the Schwartz space of G The Schwartz space is in turn defined to be the space of smooth functions f on G such that for any real r and any function g obtained from f by acting on the left or right by elements of the universal enveloping algebra of the Lie algebra of G the function 1 s rg 3 displaystyle 1 sigma r g Xi nbsp is bounded Here 3 is a certain spherical function on G invariant under left and right multiplication by K and s is the norm of the log of p where an element g of G is written as g kp for k in K and p in P References editCowling M Haagerup U Howe R Almost L2 matrix coefficients J Reine Angew Math 387 1988 97 110 Harish Chandra 1966 Discrete series for semisimple Lie groups II Explicit determination of the characters Acta Mathematica 116 1 1 111 doi 10 1007 BF02392813 ISSN 0001 5962 MR 0219666 S2CID 125806386 Knapp Anthony W Zuckerman Gregg 1976 Classification of irreducible tempered representations of semi simple Lie groups Proceedings of the National Academy of Sciences of the United States of America 73 7 2178 2180 Bibcode 1976PNAS 73 2178K doi 10 1073 pnas 73 7 2178 ISSN 0027 8424 JSTOR 65732 MR 0460545 PMC 430485 PMID 16592331 Knapp Anthony W Zuckerman Gregg J 1982 Classification of irreducible tempered representations of semisimple groups Part I Annals of Mathematics Second Series 116 2 389 455 doi 10 2307 2007066 ISSN 0003 486X JSTOR 2007066 MR 0672840 Knapp Anthony W Zuckerman Gregg J 1982 Classification of irreducible tempered representations of semisimple groups Part II Annals of Mathematics Second Series 116 3 457 501 doi 10 2307 2007019 ISSN 0003 486X JSTOR 2007019 MR 0672840 Knapp Anthony W Zuckerman Gregg J 1984 Correction Annals of Mathematics Second Series 119 3 639 doi 10 2307 2007089 ISSN 0003 486X JSTOR 2007089 MR 0744867 Knapp Anthony W Representation Theory of Semisimple Groups An Overview Based on Examples ISBN 0 691 09089 0 Wallach Nolan Real reductive groups I Pure and Applied Mathematics 132 Academic Press Inc Boston MA 1988 xx 412 pp ISBN 0 12 732960 9 Retrieved from https en wikipedia org w index php title Tempered representation amp oldid 1199303679, wikipedia, wiki, book, books, library,

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