What type of letter is Pierre Deligne talking about and why did he write a letter to other mathematicians?

I just did something like this last month...

Well, with modern technology available, I didn't write a letter; I wrote up some notes, scanned them in, and attached them to an e-mail. Keep in mind, though, that Prof. Deligne started his career in the days before e-mail, and telephone calls (from rotary dial phones attached to landlines) were expensive (like a dollar a minute.)

I had an alternate approach to proving a theorem in a paper I had read and thought perhaps the approach could also generalize beyond their theorem. I wrote up a sketch of the alternate proof (for which some details I have not actually proved) in 3 pages of notes. I scanned them, and attached them to an email to the authors that read something like "Dear X and Y, I hope you are well. We talked about this at Online Conference Z and I'm attaching some notes. I'm happy to meet by Zoom and talk. Best, -me"

I didn't write up a paper because it would take 10 pages and at least a couple weeks of work to give precise definitions of everything, and any paper would only be publishable in a fairly low-tier journal (since the result isn't new and the approach also isn't new, having been applied to other problems before), which at this point in my career doesn't count for anything.

I am a mathematician and I have written exactly this type of letter before. It was to a collaborator explaining what I thought could be the basis for our next project. As it happens, this collaborator and I are very close friends in addition to being collaborators.

In terms of format it was a letter to a friend. “Dear friend...” then four pages of math where I only had to explain things he didn’t already know and finally best wishes to him and his spouse.

Any actually time sensitive communication was already done by email and if necessary the phone. So this letter was a deliberately slow method of communication. It’s primary value was for me to sit down and think through the entire idea. That I put it into the mail is something of a secondary effect.

So to answer your “what kind of letter” it is probably best to imagine it as an informal business letter between colleagues who have known each other for a while and can include friendly chatter as well (but the chatter probably happens through other channels anyway.)

Just an oddly relevant piece of experience.

I do research in applied mathematics and computer science and, whenever I have an idea, I write an email explaining it and send it to myself.

I found that writing such emails helps me find holes in my reasoning (things I missed or skipped over) but also helps me synthesize my ideas (which sometimes leads to simplifications of the resulting concept). This emails also prove useful when I want to email other people to exchange on my ideas.

Furthermore, I write new developments as answers to the first email which gives me a centralized place to see the evolution of the idea and be able to come back to my previous positions on the subject.

The main downside is that, at the moment, I am the sender of most of my unread emails...