LET’S begin with the small matter of recognition. Have you ever, I ask Domhnall Gleeson, used your fame to your own advantage?

In his living room at home in Dublin the actor runs his hand through his hair and considers the question. “I’m sure I have,” he admits. “I know that I’ve got sat down in restaurants a couple of times. I’ve never called up and said, ‘This is who I am. Can I have a table?’ But sometimes you think, ‘They mightn’t have found that table if they hadn’t seen Star Wars.’”

Gleeson smiles. And I think, in passing, I know that smile. But that’s the nature of fame, isn’t it? We recognise the facial tics and gestures of people we’ve never met.

And, I’m afraid, as has been the case for most of the last year and a half, Gleeson and I are not meeting now. We’re on a Zoom call, me sitting in central Scotland, him across the Irish Sea. Thankfully (for me at any rate) only one of us is isolating.

Gleeson has not long returned from a shoot in upstate New York and has another day in isolation before he can go and see his mum and dad. Until then he’s stuck at home talking to me about his upcoming appearance at the Edinburgh International Festival, what it’s like coming from a family of actors and, oh yes, this whole fame thing. Which is where we came in.

“Fame scares me,” Gleeson admits. “I think I recognise as I get older just how powerful a drug it can be. In all the bad ways as well as the good ways, right? It is a drug. It can make you feel great. But you can also get addicted. And you can also run out of your supply.

“Either is bad for you. Continuing to get it or losing it.”

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Gleeson has worked with many an actor who is more famous than he is, he points out. But if he’s not famous he’s surely knocking on the door these days. Playing Bill Weasley in the film adaptations of Harry Potter and The Deathly Hallows, or General Hux in Star Wars or even popping up as Mr McGregor opposite an animated bunny voiced by James Corden in the Peter Rabbit films means your face is known at the very least. Which role does he get most recognised for?

“Depends on the age group, depends on the country. They all make me very happy,” he says.

“There was a period in Ireland, truly, when I was getting more feedback on the streets about a sketch that I wrote with a friend and did on RTE about peeing in a bottle. Or pooing in a bottle, as it turns out. I used to get stopped for that all the time and I don’t know if on nights out people want to talk to you about funny things or something, but it used to happen all the time.

“And in my head I was like, ‘This is crazy. I was in Harry Potter.’ I think it used to do the rounds on Facebook for a while. It would flare up, like herpes, and go away and then it would come back again.”

Whether you know him for Anna Karenina, in which he played Levin opposite Alicia Vikander, or Ex Machina, Brooklyn or the recent Channel 4 comedy Frank of Ireland, Harry Potter or poo in a bottle, Domhnall (it sounds like “Tonal.” But drop the T and replace it with a D) Gleeson has followed in the footsteps of his father Brendan right to the heart of Hollywood.

Indeed, he is one of those markers of quality on the big screen and the small. If you see his name on a poster you know you’re probably going to enjoy what you’re about to see.

So, the fact that he is making his debut at the Edinburgh International Festival next month, in the world premiere of Enda Walsh’s new play Medicine, a dark, absurdist piece that looks at mental health, is quite a coup.

The Herald: Gleeson with his Star Wars co-star the late Carrie FisherGleeson with his Star Wars co-star the late Carrie Fisher

Walsh is the author of the plays Disco Pigs, The Walworth Farce and Ballyturk, and the man who wrote the book for the stage version of the Irish film Once and worked with David Bowie on the musical Lazarus. Gleeson has previous with him, having performed in The Walworth Farce alongside his father Brendan and his brother Brian.

I’m presuming, I suggest to Gleeson, it was the news that the play Medicine features a giant lobster that clinched the deal for him. He has the good grace to laugh.

“The attraction was that it was Enda,” he corrects me. "I had done The Walworth Farce with my father and my brother. Maybe the best working experience of my life.

“I have always found Enda’s plays move me in a way that I don’t understand, which, when I was younger, used to frustrate me and, as I get a little bit older, I’ve become more and more enthralled by and enamoured with.

“So, when he told me he was writing this play I was very interested. I read an earlier version of it, and I was just like, ‘Yeah, sign me up.’ Little did I know that would mean a full two-and-a-half years later, because we workshopped it the Christmas before last.

“Then we were supposed to do it last summer. And so, it’s been in my blood for a long time … But I think the play is beautiful in that way Enda’s plays just affect me. I don’t know what it is. They just get me. They make me laugh and they always make me feel so … I don’t know … Melancholic or nostalgic, but in a real, beautiful way. They make me feel connected, that’s what it is.”

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Now that Gleeson thinks about it, the one time he’s been to Edinburgh during the festival was to see a mate of his in another of Walsh’s plays.

“I think I saw Penelope there. My friend Tadhg Murphy has worked with Enda a lot, and he did Penelope. Maybe that’s what I saw there.”

His memory is a bit vague. There was, he admits, alcohol taken.

“We hung out and did the full Edinburgh experience while we were there. A lot of drinks and meeting people you’ve never met before and rolling into plays you’ve never heard of, and it was great.”

That’s not what he’s expecting this time around. “No, if it was like that the play would be in tatters. I wouldn’t be fit to perform.

“So, yes, it will be different for me. But it will be different for everybody.”

But that, he says, is part of the appeal. “There is something about being part of those green shoots coming back. There’s something about being there as we grow theatre again. There’s something about that that gives it a special flavour all of its own this year. So, I’m incredibly excited.

“I don’t know if we’re going to be performing to 50 people I don’t know if we’re going to be performing to 100 people. But whoever is in there we’re going to give them all we have and there is something just so exciting about that after being away from it for so long.”

The Herald: Gleeson with fellow Irish actor Saiorse Ronan in BrooklynGleeson with fellow Irish actor Saiorse Ronan in Brooklyn

Gleeson, of course, grew up watching his dad acting on stage. When was the first time he was aware that his dad was an actor? “I don’t know. My memory is appalling. But I do remember seeing him on stage a couple of times. I saw him on stage in a Passion Machine project [the famous Dublin company of the 1980s who staged Roddy Doyle’s early writing]. I saw him play the Cowardly Lion in the Wizard of Oz.

“And” he adds, “I remember when he was in a play, and he had been offered Braveheart and they were trying to make the dates work and it looked for a while like it was not going to work. I remember the stress of that, of waiting for phone calls in the house, and realising it’s great when he’s working but it’s not great when he’s not working.

“So, I think that was probably a healthy thing to know early about the job of being an actor.”

In an interview in Interview Magazine some years ago, Angelina Jolie, his director on the film Unbroken about Allied prisoners of war held by the Japanese during the Second World War, asked Gleeson what’s it like to grow up in a healthy family? It’s a question that tells us something about Jolie’s own family story, of course. But it also tells us quite a lot about the Gleesons.

The Herald: Gleeson with his brother Brian in Frank of IrelandGleeson with his brother Brian in Frank of Ireland

“It does,” Gleeson says, laughing. “We had talked about family, myself and Angie, as I call her. And yeah, first of all, what a cool person to do that interview. That was not something she needed to do, but she did it for me which was, I think, just the sweetest thing. And she knew what a funny question that was when she asked it.

“Yeah, I’m very lucky with the family I’ve been born into. Very lucky. Just in terms of the continued support, not just in terms of the foundation. It makes your heart hurt even more when you can’t see them during a f****** pandemic, I can tell you that.”

Born in 1983, Gleeson grew up in Malahide, north of Dublin, with his dad, three brothers and mum Mary, a social worker.

“She is a very, very strong, very loving, very supportive person,” he says when I bring Mary’s name up. “She is about as good as it gets. So, by setting examples she’s taught me half the things I know, the other half being my dad.”

Gleeson started acting in school. But it was reading Martin McDonagh’s play The Lieutenant of Inishmore that really convinced him that acting was what he wanted to do.

“It just hit me at the right age. It was punky. It was like the Pogues. It was angry about aspects of the IRA, the INLA. It was understanding in some ways. It was complicated. But it was funny and critical and … It was angry young man stuff. Maybe that just got me, but it made me laugh so much. I used to read it going in and out to college every day. Nothing to do with auditions. I loved it and still do. I think it’s a wonderful play."

Eventually he got to perform it himself. “I was very good, I think, in The Lieutenant of Inishmore and Martin liked me in it. That was huge for my self-confidence. When I was doing his words, I seemed to be able to make him laugh. That meant a lot to me.”

Gleeson played the part in London and then New York, where he was nominated for a Tony. Not a bad start to your career. “And then I came home to just crickets, just nothing for like a year. So, that thing of, ‘Oh, I can do this,’ was quickly knocked out of me and I started off where everyone starts off really, which is self-doubt and self-loathing.”

The Herald: Gleeson wth Rachel McAdams in About TimeGleeson wth Rachel McAdams in About Time

Perhaps, but that hasn’t stopped him cultivating a career full of big films and big performances (he has 50 credits on IMDB, including The White House Plumbers, the TV series he has just shot in upstate New York in which he plays Watergate conspirator John Dean).

And he’s suffered for his art at times. Which, I ask him, was the harder job? Shooting for hours in the middle of nowhere in minus 25-degree weather for The Revenant or losing so much weight for Unbroken that his contact lenses didn’t fit his eyes anymore?

“I remember thinking on The Revenant, ‘This is absolutely miserable, but at least I can comfort-eat.’ Something about losing the weight on Unbroken … I was just really unhappy at that calorie deficit, so, yeah, that was definitely tough.

“And I was losing weight on my own as well. On The Revenant there were people around like Will Poulter to talk to about how terrible it was.”

Which of all of his many roles was the closest to who he really is, and which the farthest away?

“I mean a Space Nazi … I hope that is not who I am,” he begins.

“In terms of the closest? There are elements of Levin in Anna Karenina that I aspire to and there are elements of Tim in About Time that I did with Richard Curtis. I just think the love that Richard Curtis put into him … I think that’s who I would aspire to be like. Somebody who is chasing the good life. Not ‘the good life’ in quotes, but a good life. I also look very like him as well.”

That always helps, Domhnall.

The Herald: Domhnall Gleeson Photograph Sarah WealDomhnall Gleeson Photograph Sarah Weal

It’s almost time to go. Soon he will be in Edinburgh. When he does get here and, given that he’s not intending to spend the festival in the pub, what does he have in mind when he’s not onstage?

“Well, I started going indoor rock climbing when I was in Beacon [where he shot The White House Plumbers]. What do you call it? Bouldering? I’ve only gone a couple of times before, but I went there and enjoyed it. And I know it’s not a particularly Edinburgh thing, but I’ve checked and there are a couple of those sort of gyms.

“But I just want to walk around the city because it’s such a beautiful city. I want to sit outside down and have a coffee and be in the city. That’s what I want to do. I haven’t got to do it in Ireland much because when I’ve been home it’s been lockdown. I’m looking forward to just being in a city that is at least a little bit open.

“This is the green shoots. This is where it’s happening. We’ve got to enjoy it while it’s happening too. I’m very excited about it.”

So, if you see a man who looks like someone you think you saw in Star Wars, Harry Potter or Peter Rabbit then it might well be him. Hopefully he won’t be peeing in a bottle.

Medicine opens at the Traverse Theatre on Wednesday and runs until August 29. For more information, visit eif.co.uk