Timeline for What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
Current License: CC BY-SA 3.0
16 events
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| Feb 3, 2020 at 14:17 | answer | added | jorge vargas | timeline score: 2 | |
| Dec 6, 2016 at 19:37 | comment | added | Henrique Tyrrell | I am very interested in this topic and perhaps can contribute with it in the form of a question: In the same spirit as the above, what is the relation between the first picture of the principal series from induction via groups and the the principal series as summands of the coinduced rep. (à la Dixmier) of the enveloping algebra $U(\mathfrak{g})$? (for the definition and discussion of this coinduction and principal series, see chapter 7-9 of Enveloping Algebras, by Dixmier). | |
| Dec 22, 2014 at 2:17 | vote | accept | Vincent | ||
| Dec 21, 2014 at 12:50 | answer | added | Andreas Cap | timeline score: 12 | |
| Dec 6, 2014 at 5:21 | comment | added | Venkataramana | Please change "principle" to "principal" in the last two comments!! | |
| Dec 4, 2014 at 0:38 | comment | added | Venkataramana | @Vincent, Assume, for the sake of simplicity, that $G$ has a Borel subgroup $B$, with $G=KB=KAN$ the Iwasawa decomposition. If the (non-unitarily induced) principle series rep for $G$ is associated to the character $a\mapsto e^{-\chi (log a)}$, then evaluation at 1 is a linear form $w$ with character $\chi$ for the complexified Borel subalgebra $\mathfrak b$. Thus the $U(\mathfrak g)$ span of $w$ is a quotient of the Verma module $V_{\chi}$. I cannot answer the other questions since I am not at all an expert; I am sure Humphreys can tell you what the kernel from Verma to the module $W$ is. | |
| Dec 4, 2014 at 0:37 | comment | added | Venkataramana | @Vincent, firs, if we have a $U(\mathfrak g)$-module $W$ generated by a vector $w$ which is an eigenvector for the Borel subalgebra $\mathfrak b$, with eigenvalue $\chi$ say, then $W$ is a quitent of the Verma module $V_{\chi}=U(\mathfrak g)\otimes _{U(\mathfrak b)}(\chi$; so to get a subspace of the dual of a principle series representation as a quotient of a Verma module, we need to find a vector $w$ as above. | |
| Dec 3, 2014 at 17:56 | comment | added | paul garrett | @Vincent, I think there's no simple general relationship, apart from some natural dualities (when the objects are known to exist). E.g., for $p$-adic groups, the compact induction (corresponding to tensor product) is not left adjoint to restriction, although it is left adjoint to something. I saw this in P. Cartier's Corvallis notes on repns of $p$-adic groups... where some of the difficulties are illustrated. | |
| Dec 3, 2014 at 16:00 | comment | added | Vincent | @Venkataramana This sounds really interesting, can you elaborate, perhaps as an answer? In particular it seems that a subrepresentation of the restricted dual of a principal series is a quotient of a Verma. But what Verma, and how do you find the kernel of the projection? Also, is there a way of realizing the full Verma module als functionals on something? | |
| Dec 3, 2014 at 15:57 | comment | added | Vincent | @paul garett Good catch! I learned this from the notes by Vogan linked to above. Is there a general theory of how left and right adjoints to the same functor are related to each other? | |
| Dec 3, 2014 at 15:56 | comment | added | Vincent | @Jim Humphreys: Thank you for the reference, I will look them up! | |
| Dec 2, 2014 at 16:03 | comment | added | Jim Humphreys | @Vincent: You might find the older papers and lectures by T.J. Enright useful to consult. Harish-Chandra realized that real infinite-dimensional Lie group representations obtained via "induction" can translate indirectly to complex infinite-dimensional Lie algebra representations. This leads to a rigorous category equivalence between principal series for complex Lie groups and a module category for complex Lie algebras close to the BGG category $\mathcal{O}$. Of course that only deals with part of the ongoing search for unitary representations of real groups. | |
| Dec 2, 2014 at 15:15 | comment | added | paul garrett | Isn't your first construction a right adjoint, and the second a left adjoint, to suitable forgetful/restriction functors? (The analogues coincide for complex repns of finite groups.) | |
| Dec 2, 2014 at 4:15 | comment | added | Venkataramana | I am not ure what you are looking for. But, suppose you take a (not necessarily spherical) principal series representation (even the $K$-finite vectors there). The evaluation at identity of these functions on $G$ gives a linear form $\lambda$ which is "invariant" under the Borel subalgebra, and hence the $U(\mathfrak g)$ module generated by this linear form is a space of linear forms which is a $quotient$ of a Verma module. Thus there is a close connection between Verma modules and principal series representations. | |
| Dec 2, 2014 at 4:08 | history | edited | Vincent | CC BY-SA 3.0 |
corrected some spelling, inserted blockquote
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| Dec 2, 2014 at 2:16 | history | asked | Vincent | CC BY-SA 3.0 |