Compatibility of the Theta correspondence with the Whittaker functors
Vincent Lafforgue (University Paris 6), Sergey Lysenko (University, Nancy 1)
TL;DR
This paper proves that the geometric theta-lifting functor for specific classical groups is compatible with the Whittaker functors, ensuring a consistent relationship between these important tools in automorphic representation theory.
Contribution
It establishes the compatibility of the global geometric theta-lifting functor with the Whittaker normalization for certain classical group pairs.
Findings
Proves compatibility for (SO_{2n}, Sp_{2n})
Proves compatibility for (Sp_{2n}, SO_{2n+2})
Proves compatibility for (GL_{n},GL_{n+1})
Abstract
We prove that the global geometric theta-lifting functor for the pair (H, G) is compatible with the Whittaker normalization, where (H,G) is one of the pairs (SO_{2n}, Sp_{2n}), (Sp_{2n}, SO_{2n+2}) or (GL_{n},GL_{n+1}). That is, the composition of the theta-lifting functor from H to G with the Whittaker functor for G is isomorphic to the Whittaker functor for H.
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