bio | website | math.tifr.res.in/~venky |
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location | India | |
age | ||
visits | member for | 2 years, 7 months |
seen | 1 hour ago | |
stats | profile views | 2,527 |
interested in representations of groups, algebraic groups, arithmetic groups, rigidity, and related automorphic forms. Recently interested in braid groups and configuration spaces.
Dec 7 |
reviewed | Approve Roots of a polynomial in a finite field related to Fermat's Last Theorem |
Dec 7 |
answered | Is there a compactly supported function that its Fourier transfrom vanishes at given n real points? |
Dec 6 |
comment |
What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
Please change "principle" to "principal" in the last two comments!! |
Dec 4 |
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What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
@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 |
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What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
@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 2 |
reviewed | Approve Properties and name of some polynomials |
Dec 2 |
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What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
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. |
Nov 30 |
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The resolution of which conjecture/problem would advance Mathematics the most?
Grothendieck's standard conjectures? |
Nov 29 |
revised |
Action of the pure braid group on the commutator subgroup of a free group
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Nov 28 |
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Action of the pure braid group on the commutator subgroup of a free group
in short, you are asking if the kernel of the Gassner representation is non-trivial; I think this isan open problem (though it is known that the Burau rep is not faithful, and Burau and Gassner are related, to the best of my knowledge, it is not known whether Gassner is faithful. You may ask Misha Kapovitch about this |
Nov 28 |
revised |
Action of the pure braid group on the commutator subgroup of a free group
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Nov 28 |
revised |
Action of the pure braid group on the commutator subgroup of a free group
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Nov 28 |
answered | Action of the pure braid group on the commutator subgroup of a free group |
Nov 28 |
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What is the cubic casimir element of sl_3?
you may want to look at the book by Humphreys on Lie algebras. The explicit isomorphism between the associated graded of the enveloping algebra of a semi-simple Lie algebra $\mathfrak g$ and the symmetric algebra of $\mathfrak g$ is given there from which you can deduce that the space of invariants of the enveloping algebra (i.e. its centre) is isomorphic to the space of $\mathfrak g$ invariants in the symmetric algebra; the latter is a polynimial algebra with some well chosen generators. In your case, there is one in degree three, namely the determinant of a traceless $3\times 3$ matrix. |
Nov 21 |
reviewed | Approve When does the zeta function take on integer values? |
Nov 21 |
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Discrete subgroup of complex orthogonal group
If $n\geq 5$, then all lattices are "arithmetic" and classification arises from Galois cohomology , Hasse principle, etc.You have to see the books of Margulis and of Platonov-Rapinchuk for details. If $n\leq 4$, then there are many more lattices. |
Nov 21 |
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Discrete subgroup of complex orthogonal group
Do you want discrete subgroups with finite covolume or only discrete subgroups? In any case, there are many examples; if you want only discrete subgroups, a Schottky construction (ping-pong) gives you plenty. If you ask for lattices, many discrete groups can be constructed as unit groups of quadratic forms over an imaginary quadratic extension; a full classification is possible, if $n\geq 5$. |
Nov 20 |
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A good book on adeles and ideles
Weil also has a book "Adeles and Algebraic Groups" where he interprets a theorem of Siegel in terms of "Tamagawa measures" which has a fairly detailed discussion of adeles |
Nov 20 |
answered | Reductive subgroup and its derived subgroup with an irreducible represenation |
Nov 19 |
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Analytic function avoiding elements of the modular group
thank you; I missed this. |