Modular Forms course lecture notes 2

Modular Forms course lecture notes 2 - Traces Cauchy...

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( June 28, 2011 ) Traces, Cauchy identity, Schur polynomials Paul Garrett [email protected] http: / / e garrett/ 1. Example: GL 2 2. GL n ( C ) and U ( n ) 3. Decomposing holomorphic polynomials over GL n × GL n The unobvious Cauchy identity [1] is proven by taking a trace of a representation of GL n ( C ) in two different ways: one as the sum of traces of irreducible subrepresentations, the other as sum of weight spaces, that is, irreducible subrepresentations for a maximal torus. The representation whose trace we take is the space of holomorphic polynomials on n -by- n complex matrices. The holomorphic irreducibles occurring are identified by restriction to the realform U ( n ), invoking the decomposition of the biregular representation of U ( n ). Such identities arise in Rankin-Selberg integral representations of L -functions. For GL 2 , a naive, direct computation is sufficient. However, for general GL n and other higher-rank groups, direct computation is inadequate. Further, connecting local Rankin-Selberg computations to Schur functions usefully connects these computations to the Shintani-Casselman-Shalika formulas for spherical p -adic Whittaker functions. 1. Example: GL 2 First, we consider the smallest example of the Cauchy identity, namely, for GL 2 ( C ). The identity [2] X n =0 z n · a n +1 - a - ( n +1) a - a - 1 b n +1 - b - ( n +1) b - b - 1 = 1 - z 2 (1 - zab ) (1 - zab - 1 ) (1 - za - 1 b ) (1 - za - 1 b - 1 ) is easily proven by elementary algebra and summing geometric series. Slightly more generally, absorbing z into the other indeterminates, X n =0 a n +1 - c n +1 a - c b n +1 - d n +1 b - d = 1 - abcd (1 - ab ) (1 - ad ) (1 - cb ) (1 - cd ) for a, b, c, d small, or viewing everything as taking place in a formal power series ring. The above is the simplest instance of Cauchy’s identity , involving the smallest Schur polynomials a n +1 - c n +1 a - c = a n + a n - 1 c + a n - 2 c 2 + . . . a 2 c n - 2 + ac n - 1 + c n In fact, the above identities have meaning: we claim that the rearrangement X m,n 0 ( ac ) m a n +1 - c n +1 a - c ( bd ) m b n +1 - dn + 1 b - d = 1 (1 - ab ) (1 - ad ) (1 - cb ) (1 - cd ) [1] The Cauchy identity is often treated as a combinatorial result, in contexts where Schur functions are defined by formulas , rather than acknowledged to be traces of highest-weight representations . [2] This identity also arises in the non-archimedean local factors of the GL 2 Rankin-Selberg integrals: the two factors in the infinite sum are values of spherical p -adic Whittaker functions. 1
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Paul Garrett: Traces, Cauchy identity, Schur polynomials (June 28, 2011) computes the trace of the representation of GL 2 ( C ) × GL 2 ( C ) on the space C [ E ] of holomorphic polynomials on the set E of two-by-two complex matrices, in two ways. First, let g × h GL 2 ( C ) × GL 2 ( C ) act on a polynomial f on E by ( g × h ) f ( x ) = f ( g > xh ) Using transpose rather than inverse in the left action is a convenience: the function g ( g × h ) f ( x ) is thereby a polynomial function in g . On the left-hand side of the identity above, a 0 0 c -→ ( ac ) m a n +1 - c n +1 a - c is the character (that is, trace ) χ m,n of the representation π m,n = det m Sym n (std) of GL 2 , where Sym n (std) is the n th
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