Representations of SL2 and the distribution of points in ℙn

  • J. Kuttler
  • N. Wallach
Part of the Progress in Mathematics book series (PM, volume 220)


In 1958 Nagata [5] gave an ingenious argument that demonstrated the existence of counterexamples to Hilbert’s Fourteenth Problem. Recall that the original problem is the following: Let K = F(x 1, x 2,..., x n ) be the function field of affine n-space V = F n over an algebraically closed field F, and suppose LK is any subfield. Then the question is: Is A = LF[x 1 , x 2 ,..., xn] a finitely generated F-algebra. In most cases of interest, L is the field of invariants of an algebraic group G acting linearly on V, and A becomes the ring of invariant regular functions on V. Certainly, if G is reductive, the answer is yes, a result due to Hilbert himself for char(F) = 0, but much more subtle in positive characteristic (see []). More generally for G reductive O(X) G is a finitely generated F-algebra for any affine variety X over F on which G acts.


Generic Point Algebraic Group Transformation Rule Formal Power Series Hilbert Series 
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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • J. Kuttler
    • 1
  • N. Wallach
    • 2
  1. 1.Matematisches InstitutUniversität BaselBaselSwitzerland
  2. 2.Department of MathematicsUniversity of CaliforniaSan Diego La JollaUSA

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