• Source: (g,K)-module

      • In mathematics, more specifically in the representation theory of reductive Lie groups, a



      (


      g


      ,
      K
      )


      {\displaystyle ({\mathfrak {g}},K)}

      -module is an algebraic object, first introduced by Harish-Chandra, used to deal with continuous infinite-dimensional representations using algebraic techniques. Harish-Chandra showed that the study of irreducible unitary representations of a real reductive Lie group, G, could be reduced to the study of irreducible



      (


      g


      ,
      K
      )


      {\displaystyle ({\mathfrak {g}},K)}

      -modules, where





      g




      {\displaystyle {\mathfrak {g}}}

      is the Lie algebra of G and K is a maximal compact subgroup of G.


      Definition


      Let G be a real Lie group. Let





      g




      {\displaystyle {\mathfrak {g}}}

      be its Lie algebra, and K a maximal compact subgroup with Lie algebra





      k




      {\displaystyle {\mathfrak {k}}}

      . A



      (


      g


      ,
      K
      )


      {\displaystyle ({\mathfrak {g}},K)}

      -module is defined as follows: it is a vector space V that is both a Lie algebra representation of





      g




      {\displaystyle {\mathfrak {g}}}

      and a group representation of K (without regard to the topology of K) satisfying the following three conditions

      1. for any v ∈ V, k ∈ K, and X ∈





      g




      {\displaystyle {\mathfrak {g}}}





      k
      ⋅
      (
      X
      ⋅
      v
      )
      =
      (
      Ad
      ⁡
      (
      k
      )
      X
      )
      ⋅
      (
      k
      ⋅
      v
      )


      {\displaystyle k\cdot (X\cdot v)=(\operatorname {Ad} (k)X)\cdot (k\cdot v)}


      2. for any v ∈ V, Kv spans a finite-dimensional subspace of V on which the action of K is continuous
      3. for any v ∈ V and Y ∈





      k




      {\displaystyle {\mathfrak {k}}}









      (



      d

      d
      t



      exp
      ⁡
      (
      t
      Y
      )
      ⋅
      v

      )

      |


      t
      =
      0


      =
      Y
      ⋅
      v
      .


      {\displaystyle \left.\left({\frac {d}{dt}}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v.}


      In the above, the dot,



      ⋅


      {\displaystyle \cdot }

      , denotes both the action of





      g




      {\displaystyle {\mathfrak {g}}}

      on V and that of K. The notation Ad(k) denotes the adjoint action of G on





      g




      {\displaystyle {\mathfrak {g}}}

      , and Kv is the set of vectors



      k
      ⋅
      v


      {\displaystyle k\cdot v}

      as k varies over all of K.
      The first condition can be understood as follows: if G is the general linear group GL(n, R), then





      g




      {\displaystyle {\mathfrak {g}}}

      is the algebra of all n by n matrices, and the adjoint action of k on X is kXk−1; condition 1 can then be read as




      k
      X
      v
      =
      k
      X

      k

      −
      1


      k
      v
      =

      (

      k
      X

      k

      −
      1



      )

      k
      v
      .


      {\displaystyle kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv.}


      In other words, it is a compatibility requirement among the actions of K on V,





      g




      {\displaystyle {\mathfrak {g}}}

      on V, and K on





      g




      {\displaystyle {\mathfrak {g}}}

      . The third condition is also a compatibility condition, this time between the action of





      k




      {\displaystyle {\mathfrak {k}}}

      on V viewed as a sub-Lie algebra of





      g




      {\displaystyle {\mathfrak {g}}}

      and its action viewed as the differential of the action of K on V.


      Notes




      References


      Doran, Robert S.; Varadarajan, V. S., eds. (2000), The mathematical legacy of Harish-Chandra, Proceedings of Symposia in Pure Mathematics, vol. 68, AMS, ISBN 978-0-8218-1197-9, MR 1767886
      Wallach, Nolan R. (1988), Real reductive groups I, Pure and Applied Mathematics, vol. 132, Academic Press, ISBN 978-0-12-732960-4, MR 0929683

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