Langlands Duality and Invariant Differential Operators
Langlands duality is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now has seemed unrelated to the Langlands program. That is the topic of invariant differential operators. It is strange since...
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| Format: | Article |
| Language: | English |
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MDPI AG
2025-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/5/855 |
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| Summary: | Langlands duality is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now has seemed unrelated to the Langlands program. That is the topic of invariant differential operators. It is strange since both items are deeply rooted in Harish-Chandra’s representation theory of semisimple Lie groups. In this paper we start building the bridge between the two programs. We first give a short review of our method of constructing invariant differential operators. A cornerstone in our program is the induction of representations from parabolic subgroups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>=</mo><mi>M</mi><mi>A</mi><mi>N</mi></mrow></semantics></math></inline-formula> of semisimple Lie groups. The connection to the Langlands program is through the subgroup M, which other authors use in the context of the Langlands program. Next we consider the group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, which is currently prominently used via Langlands duality. In that case, we have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mi>S</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo><mo>×</mo><mi>S</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We classify the induced representations implementing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>=</mo><mi>M</mi><mi>A</mi><mi>N</mi></mrow></semantics></math></inline-formula>. We find out and classify the reducible cases. Using our procedure, we classify the invariant differential operators in this case. |
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| ISSN: | 2227-7390 |