We are trying to learn about application of geometric quantisation in representation theory, in particular, what’s called Kirillov’s orbit method.

The main reference will be Askay Venkatesh’s lectures on this, with notes taken by Tony Feng and Niccolo Ronchetti. We will put comments on this notes as we’re learning the subject, and the edited notes is here (updated 24/05/2022).

Schedule

Here is the zoom ID 872 3002 3928 with password 135105.

Talk 1 (Toan, 15/03/2022): Overview, describe irreducible representations and characters of $\text{SO}_3(\mathbb{R})$. Notes.

Talk 2 (Jaco 23/03/2022): Rewrite the character formula of $\text{SO}_3(\mathbb{R})$ as Fourier transform on the spheres (section 1.3 of Venkatesh’s notes). Notes.

Talk 3 (Toan 29/03/2022): Describe how to obtain the coadjoint orbits and define the measure in Kirrilov’s character formula. State roughly the Kirrilov’s theorem linking representations and coadjoint orbits. Notes.

Talk 4 (Toan 16/04/2022): Describe the coadjoint orbits, the symplectic forms of the orbits for the case $\text{SO}_3(\mathbb{R})$. Notes.

Talk 5 (Jaco 22/04/2022): Introduce pseudo-differential operators. Sections 3.1-3.3 of Venkatesh’s lectures. Notes.

Talk 6 (Toan 26/04/2022): Discuss geometric decomposition of $L^2(\mathbb{R})$ using pseudo-differential operators. Link this discussion with Kirrilov’s orbit method. Sections 3.4, 3.5, 4.1, 4.2 of Venkatesh’s lectures. Notes.

Talk 7 (Jaco 06/05/2022): Describe $L^2(\mathbb{R})$ as a unitary representation of the Heisenberg group. Prove Schur’s lemma for unitary representations. Sections 4.3, 4.4 of Venkatesh’s lectures. Notes.

Talk 8 (Toan 24/05/2022): Describe a character formula for the representation of the Heisenberg group on $L^2(\mathbb{R})$. Sections 4.5, 4.6, 4.7 of Venkatesh’s lectures. Notes.

Talk 9 (Jaco 09/06/2022): Describe the coadjoint orbits of the Heisenberg group. Rewrite the character formula for the representation of the Heisenberg group on $L^2(\mathbb{R})$ in the sense of Kirrilov character formula. Sections 4.8, 4.9 of Venkatesh’s lectures. Notes.

Talk 10 (Toan 15/06/2022): Classify all unitary irreducible representations of the Heisenberg group and match them with the coadjoint orbits. State the Kirrilov’s theorem for nilpotent groups. Sections 5.1, 5.2, 5.3, 7.1 of Venkatesh’s lectures. Notes.

Talk 11 (Jaco 05/07/2022): Sketch the proof of Kirrilov’s orbit theorem for nilpotent groups. Sections 7.2, 7.3 of Venkatesh’s lectures. Notes.

Resourses

Other than the main reference above, here are some more.

About geometric quantisation

Geometric quantisation’s seminar by Heidelberg University with a detailed list of topics.

Short overview on geometric quantisation by Baykara

About orbit’s method

Kirrilov’s book Lectures on the Orbit Method, with a review by David Vogan.

A talk and another one by David Vogan.