I was wondering the rationale behind naming parabolic subgroups of linear algebraic groups. The answer, interestingly, comes from the action of SL(2,\mathbb R) on the upper half plane. (I came up with this little discovery on my own.)

The orbit of the point i under the action of the standard parabolic subgroup of SL(2,\mathbb R) is a parabola.

The upper half-plane is an object that comes up in many parts of mathematics, hyperbolic geometry, complex analysis and number theory to name a few. The group SL(2,\mathbb R) acts on it by fractional linear transformations:

\begin{pmatrix} a & b \\ c & d \end{pmatrix} .\ z \mapsto \displaystyle \left( \frac{az+b}{cz+d} \right) .

A parabolic subgroup is it’s subgroup P such that quotienting by P gives a compact variety. Upto conjugation, the only parabolic subgroup of SL(2,\mathbb R) is \begin{pmatrix} * & * \\ 0 & * \end{pmatrix}. It’s action on the point i is given by:

\begin{pmatrix} x & y \\ 0 & x^{-1} \end{pmatrix} . i = \displaystyle \left( \frac{xi+y}{0i+x^{-1}} \right) = y + i x^2

whose locus is a parabola.

I wonder why textbooks in algebraic groups don’t mention this!

EDIT: (21 April, 2015) The above calculation is WRONG. I don’t know the answer to “parabolic”.

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Now I’ll think twice before going to Greyhouse. Here is why.

I went to the coffee-house today evening for a cup of GreyCap and Cogdell’s book on Automorphic L-functions. Greyhouse is a wonderful place to do some Mathematics and at times, just gaze around and see what other people are doing. My Chess buddies often meet here for some rapid games. Today the place was less crowded since the school is closed for summer.

A lady approaches me and asks me my age. I have seen her earlier around the campus and even in Greyhouse. I remember last summer, she once interrupted our Chess discussion and made an awkward conversation. Kerem pointed out she’s crazy but “not guilty until proven” is my principle.

Not wanting to create a fuss, I told her my approximate age and she left me alone. After a while, she came to me and asked if I could share the table with her. Honouring the etiquette of the coffee-house, I welcomed her. Now she asks if I could be her friend. I say yes. But soon things start getting weird.

“Would you like to be my friend with benefits?”

“No.” (What the fcuk!!)

“Do you know what that means?”

“I think I know.”

“Hugging, kissing, … ”

“Yes I know” (This was getting creepy)

“Do you have a girlfriend?”



“Because I love some things (pointing at my book) more than others.”

She laughed aloud enough for everyone around to hear. I could feel the pain hidden in that laughter. I felt pity toward her.

She tore off a page from her pocket-diary and wrote her phone number on it. When I accepted it, she asked mine. I refused to share my cell number with a creepy lady I just met. She asked her note back and I gladly obliged. With some more awkward comments, she left. I shared a moment with the Indian girl sitting at the adjacent table; I could tell she overheard our conversation.


The conversation had an impact on me I hadn’t imagined. It was not just any other crazy experience but somewhat traumatizing one. I felt shaken, mildly violated although all she did was have an extremely weird conversation. I wonder what it would be to be a girl in India. Eve-teasing, stalking, groping are real problems because of roadside Romeos – jokers who don’t know how to woo a girl but know a lot of other creepy mischief. The askew sex-ratio also doesn’t help. My (female) friend used to tell me the little ways she used to retaliate against these miscreants. They should be brought to book. Hopefully “Ab ki bar Modi sarkar” or AAP improve things.

पूजा: अाईला सांगू नकोस!

I came across a simple statement in finite group theory that I’m almost upset no one told me earlier. The source is Serre’s book ‘Linear Representations of Finite Groups’. Serre used this statement below to define / prove Brauer’s theorem on induced representations of finite groups, using which one proves the meromorphicity of Artin-L-functions. Here it goes.

G is a finite group and p is a fixed prime. An element x of G is called punipotent if x has order a power of p and pregular if it’s order is prime to p.

Cool result:  Every element x in G can be uniquely written as

x = x_u x_r;


  • x_u is p-unipotent, x_r is p-regular,
  • x_u and x_r commute and
  • they are both powers of x.

The proof is really easy. Just replace G by the (finite) cyclic group generated by x!

I call it the Jordan decomposition because we have a similar decomposition for endomorphisms (among other things).

Let V be a finite dimensional vector space over an algebraically closed field of characteristic zero (just in case!). Each x \in \text{End }(V) can be uniquely written as

x = x_s x_u;


  • x_s is semisimple (diagonalizable), x_u is nilpotent,
  • they both commute and
  • they are polynomials in x without a constant term.

Pretty cool, huh!

This post is just a collection of basic results I have compiled for referring to in desperate times Nothing too deep except Haar-von Neumann’s theorem.

A measure \mu on a locally compact (Hausdorff, always) group G is left-invariant if

\boxed{ \int_G f(s^{-1}x) \text{d} \mu(x) = \int_G f(x) \text{d} \mu(x). }

The most important theorem in this topic is the existence and uniqueness of the Haar measure (proven by Haar and von Neumann, respectively).

Theorem 1: On every locally compact group G there is a unique (up to a positive constant of proportionality) left-invariant positive measure \mu \neq 0.

Proposition 2: G, \mu as usual. For G to have a finite measure, it is necessary and sufficient that G be compact!

Proposition 3: There is a continuous group-homomorphism (called right modulus), \triangle_r : G \to \mathbb R^*_+ such that

f(xs^{-1}) \text{d} \mu(x) = \triangle_r(s) \int f(x) \text{d} \mu(x).

Proposition 4: Let \mu (respectively, \nu) be a left-invariant Haar measure on locally compact groups H (resp., K). Then the product integral on G = H \times K is left-invariant and

\boxed{\triangle^G_r(s,t) = \triangle^H_r(s) . \triangle^K_r(t).}

Corollary 5: G is unimodular (i.e., \triangle_r =1) precisely when H and K are so.

The most important example (for me) is the general linear group over number fields or p-adics and its subgroups. All semisimple (and more generally reductive) groups are unimodular. Compact groups are unimodular. Abelian groups are trivially so. However the measure on Borel (and parabolic) subgroups (i.e., upper triangular matrices) is not unimodular. The above proposition 4 allows one to transfer the Levi decomposition on the groups to their measures.

I just finished watching The Silence of The Lambs (1991).  Starring Jodie Foster and Anthony Hopkins, it is probably the first of its kind and has won many Academy Awards. The mystery thriller is rated 8.6/10 on IMDB so you can imagine how amazing it would be.

It might be just me but I find an uncanny resemblance with The Call (2013) starring Halle Berry. I watched it yesterday; the story is about a 911 call. Both movies involve psychopath killers, a kidnapped girl and a female cop. In The Silence …, Jodie Foster is an FBI agent whereas Halle Berry is a 911 operator. Both movies are centered around rescuing the victim. Both movies end with cops being misled to the wrong location whereas the actress locates the right house, barges in alone without any reinforcements, is about to be killed by the perpetrator but finally manages to overpower the bad guy!

Too much for coincidence? !!


Tailpiece: I just came across the most widely spoken constructed international auxiliary language – Esperanto. http://en.wikipedia.org/wiki/Esperanto

It has been almost a week since I returned to Purdue. The past month in India passed so quickly it just felt like a week. I was glad I could meet my family and so many friends.

I attended a workshop and conference on Galois representations at the TIFR. Situated besides the sea-shore, it always feels great to visit the Tata Institute. I was very surprised Prof. Ghate recognized me although we just met  once, that too three years ago. I met Prof. Rajan (my VSRP mentor) and my Masters advisor Prof. Anandavardhanan. It was nice to finally meet Sandeep Verma, a student of my advisor Dr. Shahidi. I couldn’t meet Anand Sawant since he was making an academic visit to Germany but it was good to talk to Sachin Sharma, my old friend at IMSc doing his post-doc at Tata. Although I couldn’t meet Arghya, I made friends with his friends – Shaunak, Aditya, Vineet and Ashay – and we discussed some Mathematics. There were prominent mathematicians attending the conference including Marie-France Vigneras, Sujatha Ramadorai, Pierre Colmez, Dipendra Prasad and Chandrashekhar Khare.


Conference photo

Back home, it was after a long three-year wait that my parents, sister and I were together. We celebrated by going on a short vacation to Jaisalmer and Jodhpur in Rajasthan. There were sand dunes around (it’s a freaking desert) and we had a camel safari. It was fun!

The vibrant colours of India

The vibrant colours of India (cloth shop inside Jodhpur fort)

Rajasthani folk dancer (Yes, they are glasses she's balancing on)

Rajasthani folk dancer (Yes, they are glasses she’s balancing on)

On the pretext of giving his iPad to his folks, I met Partha’s dad.  Fortunately he didn’t ask me usual questions Indian parents ask! Shashank is one of my best friends and he too came to India to attend Avdhut’s marriage. We went to visit Vaibhav (photo below) and had an interesting discussion on genetics and evolution. As always, it was fun meeting Amar – my roomie at IITB. He went paragliding near Pune a day before I left for the US and seeing these pics, I regret not being able to join. Yes, I also met my Purdue roommate Nikhil, who came all the way from Pune to meet me. Nikhil, Rubin and I had a great time cracking Purdue jokes here in India. Another close friend Ameya married last year and in this India visit, I met Asmita and him. (Everyone seems to be getting married :( I’m getting older.)

Vaibhav and Shashank

Vaibhav and Shashank

The star of the show was Avdhut, whose marriage was a pushing reason I visited India. His brothers had arranged a bachelors’ party a week before the wedding and I won’t go into details because what happens in Murud stays in Murud. But yes, I did meet Arun and have curious philosophical discussions with him. Avdhut’s marriage was a good excuse for VNITians to meet up – Raju, Tapa, Swapnil, Milind, SKS, Khan and the great Maythegod! Bless the newly wed couple!

Avdhut weds Aasawari

Avdhut weds Aasawari

My awesome dream-visit was brought down to earth after landing at Chicago. It was -27 degree Celcius cold!


* Just yesterday, I got a Macbook Pro! I promised myself I’d write this post on my new Mac.

* I was stuck up in Chicago since the interstate I-65 was dysfunctional due to heavy snowfall; shuttles were canceled and the train was overbooked. Kyle rescued me by giving a ride since he too was driving from Chicago back to WL. Hail Facebook!

Kyle gave me a ride to Purdue

Kyle gave me a ride to Purdue

* Steve Spallone is visiting Purdue next week. He is an American mathematician working in India and I’m quite the opposite. We discovered each others’ blogs accidently. Curiously, he too works in number theory and in fact is Shahidi’s collaborator. I’m looking forward to his talk!

The land of organized chaos

The land of organized chaos

Let G = GL_2(F) for some field F and B be the subgroup of upper triangular matrices in G. As usual, let \mathbb P^1 = \mathbb P^1(F) denote the projective plane over F.

As F-varieties, G/B is isomorphic to \mathbb P^1 because B is the stabilizer of the usual (Mobius) action of G on \mathbb P^1. The resulting action of G/B on \mathbb P^1 is 1-transitive and fixed-point free. Indeed, counting points when F is a finite field with q elements agrees with the facts that

\# G(\mathbb F_q) = (q^2-1)(q^2-q)


\# \mathbb P^1(\mathbb F_q) = q+1.

This generalizes in that parabolic subgroups (closed subgroup containing a Borel subgroup, i.e., containing the upper triangular matrices upto conjugation, for the general linear group) of an algebraic group give rise to projective varieties. They are precisely the stabilizers of a flag and conversely, the set of flags can be given the structure of an F-space and in particular a topology wherein they are compact spaces (being projective spaces). They are called flag varieties.

It is but with a heavy heart that I take to pen down my feelings. Sachin batted in what is very likely his last international innings today. No more Sachin. And to make matters worse, Anand lost the fifth game to Carlsen after drawing the last four games.

They measure life by the moments that take your breath away. I can reminisce my life so far by the Sachin moments I can recount. He was always there all my life. I distinctly remember enjoying what would be known as the Desert Storm, watching on my our old TV, in seventh grade. Then there was Sachin’s 98 in the India-Pakistan match in the 2003 world cup, two days before my Maths exam in HSC. And how can I forget the first one-day double century ever by Sachin; frenzied IIT-B hostel-junta was all jumping on the mess tables. And there was the World Cup India won, in 2011, when Virat Kohli carried Sachin on his shoulders. Those Sachin moments I associate with the good times I had studying Mathematics at IMSc. With Sachin retired, there won’t be any more of these moments. Dhoni puts it succinctly – “With Sachin Paaji, a part of me will be gone too.”

I respected, loved, adored, worshipped Sachin but frankly, I never wanted to be a cricketer myself. But at least for a little while as a kid, I wanted to be a grandmaster. Like Sachin, I grew up seeing Vishwanathan Anand. Chess was amazing and Vishy kept winning those rapid and blindfold games and I just idolized him. He became the World Champion in 2007 and has retained the title until now. After four consecutive draws, finally blood is drawn he loses to Magnus Carlsen. With seven more matches to go, I am badly hoping Anand resurges like a Phoenix in the remaining matches and clinches the title. It doesn’t matter if Carlsen snatches the title back from him next year. It might sound silly but, for me it’s like a battle between Sherlock Holmes and Professor Moriarty; it would be the crowning achievement of Holmes’ career if he could defeat Moriarty. Not that I hate Magnus Carlsen, but it’s just that it would pain me to witness Anand fall.

I believe it’s not just the fact that they have been the best players in their respective sports that puts Anand and Sachin on the same pedestal. Of course, both are Indians and I have grown up worshiping them both. But the real similarity between them is that both have their feet firmly on the ground. They both are the epitome of humbleness, a quality too difficult to exaggerate. It comes only with immense patience and respect towards your surroundings. In 23 years of his professional career, I don’t recall a single outspoken statement Sachin made, nor the slightest harsh action on the field. Hit by a speedy bouncer, he would calmly pick himself up and continue to bat in pain. And Anand, well, I’m honoured to see him nay, play a game in an exhibition match with him. It was at the the ICM at Hyderabad in 2010. People asked random, stupid questions to him and he answered them all with aplomb. Take a bow, masters!

Why do we need idols?

My sadness at Sachin retiring and Anand losing a game to Carlsen got me thinking; why did I feel sad? Because I wouldn’t be witnessing my role models in action. And why do I need role models? I study Mathematics, neither of them are anyway remotely related to Mathematics. That’s not the reason I idolize them. I revere them because seeing them in action makes me believe I can excel in my field too. I equate myself with them. I want to be a part of their success, and even failure. If Sachin can persevere and perfect that straight drive, I too can push my cognitive limits and understand my Mathematics. If Anand can produce a masterpiece, may be one day I too can.

I hope Anand fights back tomorrow and comes up with something brilliant. He always has.


An article I wrote about Sachin long ago — here.

A moment to cherish:

Vishwanathan Anand and me

In my recent Riemann surfaces class, Donu started talking about de Rham cohomology and it’s generalization to Hodge theory. I was fascinated by it (in particular, the Hodge decomposition) so started reading some related stuff. I initially intended to write a blog post about it, but soon it grew in size than I had intended, so I’m attaching it to this post as a pdf file. Talking to Nick and Partha and looking at Forster’s Lectures on Riemann Surfaces was also helpful.

A mild introduction to Hodge_theory.

About me

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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