## Mirković–Vilonen basis in type $A_1$

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- by Pierre Baumann and Arnaud Demarais
- Represent. Theory
**25**(2021), 780-806 - DOI: https://doi.org/10.1090/ert/582
- Published electronically: September 29, 2021
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## Abstract:

Let $G$ be a connected reductive algebraic group over $\mathbb C$. Through the geometric Satake equivalence, the fundamental classes of the Mirković–Vilonen cycles define a basis in each tensor product $V(\lambda _1)\otimes \cdots \otimes V(\lambda _r)$ of irreducible representations of $G$. We compute this basis in the case $G=\mathrm {SL}_2(\mathbb C)$ and conclude that in this case it coincides with the dual canonical basis at $q=1$.## References

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## Bibliographic Information

**Pierre Baumann**- Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
- MR Author ID: 633326
- ORCID: 0000-0002-6947-0778
- Email: p.baumann@unistra.fr
**Arnaud Demarais**- Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
- Address at time of publication: 15 allée du puits, 01290 Crottet, France
- Email: arnaud.demarais@ac-dijon.fr
- Received by editor(s): December 21, 2020
- Received by editor(s) in revised form: April 15, 2021, and May 24, 2021
- Published electronically: September 29, 2021
- Additional Notes: The first author was supported by the ANR (project GeoLie ANR-15-CE40-0012).
- © Copyright 2021 American Mathematical Society
- Journal: Represent. Theory
**25**(2021), 780-806 - MSC (2020): Primary 22E46; Secondary 14M15
- DOI: https://doi.org/10.1090/ert/582
- MathSciNet review: 4319510