Timeline for A question on representation theory of p-adic groups
Current License: CC BY-SA 4.0
10 events
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| Nov 10, 2021 at 18:17 | history | edited | Alex B. | CC BY-SA 4.0 |
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| Nov 10, 2021 at 17:15 | history | edited | Joël | CC BY-SA 4.0 |
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| Sep 26, 2017 at 1:05 | comment | added | JACK | Thanks to both of you, Paul and Joël. @Paul Broussous On the book "The local Langlands conjecture for $GL(2)$" you just mentioned, I found the chapter 3 is mainly about induced representations. But in the whole chapter, I did not find the corresponding proof. Maybe I missed something or I searched for the wrong chapter. Could you kindly tell me where can I find the theorem and proof? Many thanks! | |
| Sep 25, 2017 at 14:04 | comment | added | Joël | "sum of values") is irreducible. Still doable. The trick is to show that any non-zero sub-rep in $V$ has a representative in $W$ with at most 2 neighborhing points as its support. Anyway, like this you can get a complete and elementary prove that V is indecomposable, and has 2 Jordan-Holder factor, the trivial representation and $V_0$, which is called the Steinberg. | |
| Sep 25, 2017 at 13:52 | comment | added | Joël | Paul's reference is good, or if you are ready to dip directly into the general theory (not only for $GL_2$, you can try the article of Cartier in Corvallis (ams.org/books/pspum/033.1). But this is why I also gave the second example with the tree. For this example, determining the structure of V is a series of really doable exercises. You need to prove that V has the trivial representation as quotient: done. That V has not the trivial representation as sub-object. Really it is not hard. And finally to prove that the kernel $V_0$ of the map $V$-to-trivial (that is the map... | |
| Sep 25, 2017 at 9:28 | comment | added | Paul Broussous | A good reference for beginners is the monography by Bushnell and Henniart "The local Langlands conjecture for ${\rm GL}(2)$. They give a complete proof that the (admissible) induced representation from the trivial character of the Borel subgroup has length $2$ and is indecomposable. It has the trivial character as subrepresentation and the Steinberg representation as quotient. | |
| Sep 25, 2017 at 7:02 | comment | added | JACK | Thank you very much! But you know I'm a beginner of automorphic forms and representation theory. So I have a silly question: why the principal series which are induced from a character of the Borel subgroup of $G$ are always indecomposable? Could you briefly give a proof or give me some references to read? Thank you! | |
| Sep 24, 2017 at 19:01 | history | edited | Joël | CC BY-SA 3.0 |
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| Sep 24, 2017 at 18:19 | history | edited | Joël | CC BY-SA 3.0 |
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| Sep 24, 2017 at 18:13 | history | answered | Joël | CC BY-SA 3.0 |