Relative Kazhdan Lusztig isomorphism for $GL_{2n}/Sp_{2n}$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914295061676032 |
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| author | Shtotland, Guy |
| author_facet | Shtotland, Guy |
| contents | The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a $p$-adic group to the equivariant $K$ theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne Langlands conjectures concerning the classification of smooth irreducible representations with an Iwahori fixed vector. In this work we state and prove a relative version of the Kazhdan Lusztig isomorphism for the symmetric pair $(GL_{2n},Sp_{2n})$. The relative isomorphism is an isomorphism between the module of compactly supported Iwahori invariant functions on $X=GL_{2n}/Sp_{2n}$ and another module over the affine Hecke algebra constructed using equivariant $K$ theory and the relative Langlands duality. We use this isomorphism to give a new proof of a condition on $X$ distinguished representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22846 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relative Kazhdan Lusztig isomorphism for $GL_{2n}/Sp_{2n}$ Shtotland, Guy Representation Theory 22E50 The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a $p$-adic group to the equivariant $K$ theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne Langlands conjectures concerning the classification of smooth irreducible representations with an Iwahori fixed vector. In this work we state and prove a relative version of the Kazhdan Lusztig isomorphism for the symmetric pair $(GL_{2n},Sp_{2n})$. The relative isomorphism is an isomorphism between the module of compactly supported Iwahori invariant functions on $X=GL_{2n}/Sp_{2n}$ and another module over the affine Hecke algebra constructed using equivariant $K$ theory and the relative Langlands duality. We use this isomorphism to give a new proof of a condition on $X$ distinguished representations. |
| title | Relative Kazhdan Lusztig isomorphism for $GL_{2n}/Sp_{2n}$ |
| topic | Representation Theory 22E50 |
| url | https://arxiv.org/abs/2601.22846 |